Lots and Lots of Ramanujan-Style Congruence miracles and Lots and Lots

                     of new Integer Sequences for the OEIS



By John Chiarelli, Rebecca Coulson, Bryan Ek, Keith Frankston, Alejandro Ginory,
Emily Kukura, Andrew Lohr, Jinyoung Park, Ali Rostami, Xukai Yan, Mingjia Yang,
and Anthony Zaleski



      Recall that p(n) is the number of integer-patitions of the integer n.

       Leohanrd Euler famously proved that these are the coefficients of



                                infinity
                                --------'
                               '  |  |      1
                                  |  |    ------         .
                                  |  |         i
                                  |  |    1 - q
                                 i = 1



                   Srinivasa Ramanujan famously proved that

                1/5 p(5 n + 4), 1/7 p(7 n + 5), 1/11 p(11 n + 6)

       are all integers. These integer sequences are already in the OEIS.

         These amazing facts also led to the question, first raised by

     Freeman Dyson, to give combinatorial explanations. This was answered

                     by Frank Garvan, and George Andrews.



  In this article we will state many new such congruences for the more general

                             sequences defined by



                               infinity
                               --------'       i r
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i s
                                 |  |    (1 - q )
                                i = 1



for r from, 1,  to , 17, and for s from, 0, to , 17,except the trivial case r=s=0,
and also supply the first, 24, terms of many new sequences for the OEIS.



  Each theorem also raises an intriguing challenge to prove it combinatorially
by finding a way to partition the counted set into subsets with the same cardinality.



                                                    n
       Let's call, c[r, s](n), the coefficient of, q , in the above infinite product.



     Note that these also have natural combinatorial meanings. For example

              The case r=1, s=1 counts so-called  overpartitions.



    We don't bother with the proofs, since for the case r=0 it was shown by

            Dennis Eichhorn and Ken  Ono, in their interesting paper

                       "Congruences for partition functions",

that appeared in

"Proceedings for a Conference in Honor of Heini Halbertstam 1, 1996, pp, 309-321",
available from 

http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/013.pdf ,

that one can easily determine an N_0 such that the theorem is true if it is  checked
for n<=N0. It turns out that the N_0 are usually rather small, and since we check it
for many values of n, and we know that there exists a rigorous proof, we don't bother
to actually prove it.

                                                                       n
        Theorem Number , 1,  Let , c[1, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[1, 4](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 211, 9159, 207052, 3170310, 37127511, 356116878, 2921227999, 21105373304,

    137167587602, 814705215526, 4476425074102, 22973625421812, 110987787807775,

    507978630301959, 2214398534103009, 9235519680550783, 36993985081846070,

    142791934258485535, 532634787201341833, 1924881461300739174,

    6754496733833214225, 23059697883960100469, 76727411996280352911

                                                                       n
        Theorem Number , 2,  Let , c[1, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[1, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12, 1062, 33801, 640747, 8719909, 93676890, 840457319, 6533359870, 45146748535,

    282565239561, 1624709220984, 8677641890801, 43434099980726, 205201992704635,

    920517630854659, 3940427835096784, 16163980921423278, 63770226221412603,

    242723101822675360, 893743158829473441, 3191270418185088791,

    11073474602704401837, 37410368531947551647, 123259242312798112260

                                                                       n
        Theorem Number , 3,  Let , c[1, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[1, 6](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 3899, 1031295, 98758436, 5232966637, 185475274004, 4884366526843,

    102048213463581, 1768157561923317, 26223691770616418, 340838205223338387,

    3953140630865203635, 41502386518449919007, 398963019918620034861,

    3544978891743697047636, 29344186723039024577838, 227787064782591567080998,

    1667562193364363488278407, 11568786069211984921480635,

    76379279362052302457402944, 481665889423968800196264105,

    2910776969177476943230850193, 16904984821000135064402858224,

    94597130647647973641314684281

                                                                       n
        Theorem Number , 4,  Let , c[1, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[1, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

200, 111568, 15276710, 1011126294, 41892090401, 1240830852493, 28445337499467,

    531639462845895, 8400654239392161, 115253009146465924, 1400878619109334426,

    15324964309607638826, 152796404510057739608, 1402778298125503551241,

    11959303445378274488137, 95356082847551995861914, 715375465134821752581879,

    5075827248152507398319349, 34214346579000623928305930,

    219953880432396575054874021, 1353208065512894305847775448,

    7991393033143021247759718731, 45422948667594248119517238332,

    249097424976179377480054718085

                                                                       n
        Theorem Number , 5,  Let , c[1, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[1, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

571, 239572, 28833402, 1764384767, 69230965267, 1968930405453, 43719277332291,

    796189679452364, 12311670037274799, 165828364127393528, 1983761890531498170,

    21400597995760569268, 210750105032857369679, 1913549965393452516891,

    16152003803294555273943, 127625871384482295721342, 949589114170863069691854,

    6686738525816846658435573, 44758711487775719541062478,

    285882134108789572638481828, 1748251039008242449586176869,

    10266467218146734728122635030, 58048333293457297522928384459,

    316767038858708538555044239315

                                                                       n
        Theorem Number , 6,  Let , c[1, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[1, 7](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

26700, 400353456, 673506889781, 368729649688032, 98315825053995322,

    15688826422239296192, 1693742211499960976517, 134021848071406662592624,

    8218833066387936693168769, 406866673295694500383526312,

    16768932553334886862994693583, 589405169030958811726368291248,

    18009677736868629904827614548458, 485904940595152145198297713860192,

    11725810504683956791887644382648177, 255838108116323026065974844173298480,

    5093184572018822663771127138840250785,

    93243111413418572667485811074644315504,

    1580474568063994326192599490521604496290,

    24949630527820603528342883273175264900576,

    368719458452111421962991067713486335467849,

    5124730458315908131174970756101045280346816,

    67259840030091664416689990211371902117374061,

    836622165259159427740107073438173393831939744

                                                                       n
        Theorem Number , 7,  Let , c[1, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[1, 9](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 3584, 737024, 61436162, 2977715836, 99231526528, 2499785127936,

    50550828023016, 854926923859004, 12454223172591104, 159766774572276480,

    1835911662111071211, 19155124762340525608, 183456986104506024128,

    1627440729440625440128, 13472786299978319140764, 104748017527525823440198,

    768995915372973709054976, 5355774979597580656918528,

    35530947146497325795394641, 225333295096185320380879452,

    1370389426025414025326458432, 8014480077889589962681545216,

    45185803347714068068119333940

                                                                       n
        Theorem Number , 8,  Let , c[1, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[1, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

64, 35013, 4689304, 306028096, 12581941248, 371561709700, 8522108328318,

    159778385152000, 2537848941323264, 35055399584107799, 429547338578408180,

    4742085188135218240, 47754203603167410176, 443119378449898761372,

    3820533379412205798596, 30822267405093364820992, 234059402752046501387136,

    1681607629865991599640552, 11480971066062790222440960,

    74775942067159977717208000, 466171526359668828887758592,

    2790176622968587634581844776, 16075995424318156878917442414,

    89375872409409147490075852800

                                                                       n
        Theorem Number , 9,  Let , c[1, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i
                              '  |  |     1 + q
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



              Then , 1/13 c[1, 9](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

38796, 2370593280, 10821383256448, 13482845993887615, 7367355677823279080,

    2242431323434603705088, 437825188838293829079040,

    60104313114353302026904744, 6183075163199291107275939390,

    499295067281197831096250754560, 32779352265779179263128969386496,

    1797991426474786404505931061868170, 84209312174955569967487419974578296,

    3427557371656071313439767155736001920,

    123024048900098707884897496292054824960,

    3941605484057141341580613035117198378832,

    113899473379479865450177258267409594584900,

    2994866889286846144420391575631371559822848,

    72203807954050677476272172044538522200167680,

    1606798031030738016504240663651752727361918705,

    33198530500695029549272012115388814688483896744,

    640139408192992433052536958481725207536516805824,

    11572321014417724003556536185355866052349935120896,

    196941103221043850668060050322994900199309043098840

                                                                        n
       Theorem Number , 10,  Let , c[1, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[1, 10](11 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 1728610, 17363579061, 30077171659162, 19158273900890113, 6232514071257936628,

    1238597736727736897577, 167976688725115085493198,

    16744694898844674900931389, 1293381402394578529329205613,

    80496874167481518129582564855, 4159526203963227090413582061430,

    182706390063621888825170211616252, 6952365300715827150337558410445637,

    232760837452588479065433302720378727, 6945072727795482686526429931409011068,

    186697936116492737937136547506360938264,

    4563569504399208594213835075901904246782,

    102239141764645976785572986562389141924199,

    2113806182500964644003611823481949549820842,

    40575197040850654287608036449311625248928635,

    726947380465442085412161927882192305279284668,

    12213187762823773851272216063180566084281704613,

    193220601523180371914467160335646265092324152433

                                                                        n
       Theorem Number , 11,  Let , c[1, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[1, 10](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

675, 66078961, 303099453551, 345622501359293, 167768996475256408,

    44893560977802778792, 7683655907293440956880, 924972569396076778576595,

    83596680984704031223241058, 5945752327761065528862028835,

    344780812372510787384403476784, 16752771769258608242048307599765,

    697054791462297631528881390321598, 25276370289591884472470175630921305,

    810421018373437301648960048782540836,

    23254073333827178817052280754264488776,

    603271774420381922828735135069613576206,

    14273845248180965267802263322975940080365,

    310349517538960310614514854622890699017946,

    6241461541421689832609288355048467100256013,

    116771928961268370472058714997126668898722143,

    2042707656971732297444053006593036816310477715,

    33561503797176365936760695689996835650313661839,

    519979571333879321733949657126434135511378597719

                                                                        n
       Theorem Number , 12,  Let , c[1, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[1, 10](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2471, 154670703, 601837016922, 625048912668588, 285018595294754177,

    72860938093606445120, 12037228475361466112877, 1408458178477829530071396,

    124334673417254027459440800, 8669115672402701004472005243,

    494175870501514335029811109037, 23656065446917659009688069432528,

    971404748769962022194600897465998, 34813423718888557338934699916923745,

    1104478076198654769840784349492896048,

    31390157510851628785067839799169639860,

    807280454379640861000031465947910325327,

    18949003515300326609211612757320022366176,

    408982953241398553413817954463279327297691,

    8169382649465314904910462152620065901433252,

    151880292117747353597108169710899529656945379,

    2641290914015449017612451733598239071916067766,

    43158369594844711667286773337207298281341906962,

    665231160006060005555952823146078908268720991982

                                                                        n
       Theorem Number , 13,  Let , c[1, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[1, 10](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8387, 354732237, 1182569182562, 1122884398186615, 481931116473860691,

    117829126604498961988, 18804206096008840635939, 2139699162320708988830001,

    184566094271654365116597785, 12618917056348805051774920153,

    707285509541449706200287769087, 33361770596116265723742834528431,

    1352215449373566509405106975392455, 47900710851824731158877449544113753,

    1503866253629030587748520618688130135,

    42337933607175225774919251105483013769,

    1079464519246446585238428469206274057560,

    25138008536121721551588232104685665276291,

    538618913652841698386416709624736333588929,

    10686481934372630124990034987689567336835900,

    197435348517127918452235265602083169881301849,

    3413522048138259564389384854952582135177111408,

    55472704028447661774418493521980446283288071568,

    850672416951793670156971657251070851232481800539

                                                                        n
       Theorem Number , 14,  Let , c[1, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[1, 10](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

80797, 1765311208, 4433151626164, 3555453618311919, 1359188528608146224,

    304964086401037894258, 45510926223222948349514, 4904767231021709882943820,

    404380869181035151874172965, 26606823472432423122479009456,

    1442699433972792384742759208166, 66105695105900446227959135150667,

    2611525927471638806740541894398166, 90414614102825424377235951616003471,

    2780638928506950403192544170264732921,

    76831453151764355058233687650869598418,

    1925768289755032552078779623954262151492,

    44149529503682681371016165728486745083760,

    932414400203463276985315235479256801492040,

    18254098728296815033205948704240547740392637,

    333088414602714168172689547827716807172588453,

    5692597172470146663683126619861560855798092257,

    91513439052785908000220734765064814097183127988,

    1389174033506605206584323605695574852313506352842

                                                                        n
       Theorem Number , 15,  Let , c[1, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 11](17 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

95506525, 39219469477656, 1066034477214355503, 6685047782403478306384,

    16040848077726556679048293, 19228171621631933471355961768,

    13531858316936266637017717995727, 6220767144089779229221084447703360,

    2013604189024317804504856555983181192,

    484947294938263725653911359615089893148,

    90621092553256914738544317638693964810565,

    13577974473846752943300152072382208085302928,

    1674625878541357057481522142547089025781067440,

    173690148929822994704398501105604843055353983080,

    15420769811085093789797729641337167502170164128291,

    1189520934121244207209359020827286275350831200530480,

    80734701153830184086845517420431546165180661470593889,

    4873910282079120944770302484442078713201061355390412976,

    264179060611496994012822066835482828880706642207622205117,

    12962178433931772357881884003235974268942127489672553729744,

    579883407236532852944484054803257965174490771447525355496313,

    23803997205908054112720968730570101988757760014970643772848112,

    901704249151205588138913568333448365312701999859713272287561287,

    31679780384994215056616292324368156750125401619248763713674950896

                                                                        n
       Theorem Number , 16,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48, 265471248, 10657976995616, 62808267394333619, 120241692179006341205,

    107035203459483524456106, 54079873840543728440250609,

    17570805446691954962113734669, 3994466349904022246786274116943,

    674894047894326752902695129848314, 88639928772794479971967045103260478,

    9367809859641701670117353466025417991,

    818639202182331455497299697160558246938,

    60467795926483234765997682867530573679714,

    3843685405145812699102026949691938244317281,

    213435969741200261257536716431726398689429553,

    10484946396494689600782213899404022413482672370,

    460592894476407767897315173402731786229014272997,

    18261800485544104371017345647316955059383257507544,

    658776570842388383872120795430318535612494617874790,

    21774936689027502637940837468005986752387137861275365,

    663578252425404263128916816157221060553597622320026450,

    18746944353197815583919682836468650055650218185077103431,

    493400428596669595294243205941585390417925475741571602300

                                                                        n
       Theorem Number , 17,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4409, 3931542901, 89937698527142, 390298779763775623, 611414285548617281542,

    470862858442719598740328, 212882600124815108302319268,

    63287083740957249534108044741, 13372185248710214084121609761570,

    2124208806924929431096776008655859, 264613089048228017332615145810420559,

    26706325724880527768765915884771229264,

    2240998436448551433675191723206595202898,

    159655842611172533182666611135920695291127,

    9824837431509622355041093360322972172103196,

    529796563596184743131211270979236163557342786,

    25340522557509378416279772685676696790446472100,

    1086314605301162254233721798712372159921851832072,

    42113402091336810325055730809434715018598250574173,

    1487970390854531622779480368067850463343338542882391,

    48244044469222773929721481745620176127123234172965311,

    1444061347956997246669836716311210029031927902578948692,

    40118357927588811508395763142045384676405431708879536492,

    1039416683093067429756885177771666667497620574471356205962

                                                                        n
       Theorem Number , 18,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

16455, 9279003519, 179592743913917, 708974120691415114, 1042566798695211805794,

    766658119697062343572642, 334459539321021374911632567,

    96618335057649668537239448460, 19936262261778941249099767919898,

    3104052995662729695938952076522263, 380059853725850196335218852297545432,

    37785044293614755112466326146365582085,

    3128814859891892016158336943461135247982,

    220283879889148171219683245880267958194355,

    13412298660109244668490856950190733536781792,

    716315673618908486078783004335393026092897487,

    33962565275404568929082031923375400435614687255,

    1444274782008763886935613442485868117966163029465,

    55577742478468636739561308251226042711549576356590,

    1950308304549001693591482418324469774194137543036120,

    62833901073838918873010655473532223850726473000422523,

    1869673629421625739984481022337279823481189950991738001,

    51656027446831420968732988044577283359026772913303223281,

    1331421225357472716001735930845775367857271312327207509527

                                                                        n
       Theorem Number , 19,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

57437, 21516511673, 355402874073521, 1280504363601829856,

    1770578066334715427033, 1244455717518583829070956,

    524197109494275449970853076, 147212137121896275224391184489,

    29673190356978935560410186727555, 4529449811640109443552776846002157,

    545203721961876185215362632730437335,

    53401920013372403461444467262717798114,

    4364173769467544019114497179068329864491,

    303674228795798134272883239636843850482963,

    18295518357012544667946902288013961715230333,

    967819103604672584472957254317830301599512761,

    45488973481267411897666480272626074054309301537,

    1919055285406924606568276784932197106617121398924,

    73306874559955596593553169291094909900136204691438,

    2555011289182769073372284607433643942452297803511214,

    81797536853394571760819391863203503275092205921282703,

    2419666162656862012725882961427530921112021432529368519,

    66484513740950495000422199817865054014699231907543366951,

    1704801459011614621255796674710147504778432941754515233792

                                                                        n
       Theorem Number , 20,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

189398, 49074340257, 697259084488672, 2299968792632823436,

    2995090055875858579447, 2013970387613031045271501,

    819616162584336964555676418, 223861592909072958942893239729,

    44093344421860965420390610542232, 6600140726905616327745298492494935,

    781155139321434896798802080534725502,

    75392717219084493302788808748921243683,

    6081513860096346907311702726244625805596,

    418276535374513135162605312771203991966212,

    24937472254714653574485620838998524015278226,

    1306713913708424160296082759199284342547937061,

    60888374940943488079412981468202964960263953494,

    2548415344683222024710301923115865709039510069459,

    96639160091691662756645276268379380446585824326177,

    3345524348722192101049620182469601940885797685138424,

    106434743782270747299617593165227894411297867711596210,

    3130080973486303435184422079884177702297159277804241515,

    85534733560412827124473532464849776388304762710320064305,

    2182053889838404279239502508335124026267060453696213506282

                                                                        n
       Theorem Number , 21,  Let , c[1, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[1, 12](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5172585, 532202987361, 5015008124450538, 12911041522389490914,

    14170340240502846449232, 8388662727954681348470074,

    3089675119518394295317158224, 778247259251100543490904164639,

    143290618182247687527426745983687, 20253175742610358023495541015705379,

    2281164419681249723411210312903131797,

    210816161901017079043419230082255333977,

    16364445362488049478640092759107105877673,

    1087531663635677273443091121095922516047555,

    62863391702645785763997973427336032326817616,

    3202877546701721757195268580172564901613897032,

    145469869162837133459657735055490224424406914369,

    5947092846672351042005910546427846114803829482013,

    220688784676621690140839125414364047658376474315658,

    7488270082845820652775772684300889817135347126652232,

    233833014795424365392848558316123881472525199906677146,

    6758160143896901006583388030897420317367546177657914964,

    181698917036634069642893454418099933578563527099835659590,

    4565078818725025155200848535069692022883127139018893353936

                                                                        n
       Theorem Number , 22,  Let , c[1, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[1, 13](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 162759, 378118316, 222745348578, 58557950604256, 8868335418056203,

    892044083721145704, 65163953430361184994, 3672856850271823723568,

    166805290592261111247406, 6304877996683184962583472,

    203342415804359248999505368, 5707104808114917191581489310,

    141625233989852013825889083069, 3148241327253007365882294727260,

    63375189242364974462855584313410, 1165947780727965381734082627320858,

    19758223063472159088390206328250192, 310495704089353993002979674518674376,

    4551437543730298189460325803634107724,

    62554239843654433349469886070629237920,

    809741192019897364289807967825124349218,

    9912001040907696853810327197869913583784,

    115148484229841732868135319485099030460264

                                                                        n
       Theorem Number , 23,  Let , c[1, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[1, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2710, 17070590, 16329283580, 5817512510556, 1091712553774184,

    129139936252262416, 10725693887408268488, 671471291502079558166,

    33302320978517934509254, 1357152224976528538597830,

    46725350902557672996421124, 1388930677328246534178092094,

    36271052587958326327107442704, 844003946522368450351411751710,

    17706391372225940757003685525088, 338217352231624896857794549459968,

    5931668589275036900624617127766592, 96203884968621600860119592902998946,

    1451926444675510505814663190906507644,

    20501763106730930813011428138460033726,

    272148462566172889173799759043550905506,

    3410542488578990290120685037974818133976,

    40502054801822795299210063194212756433524,

    457328149620486196786315028492805213988646

                                                                        n
       Theorem Number , 24,  Let , c[1, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[1, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11474, 49445902, 39718240648, 12712691597284, 2214036464063038,

    247633263930376386, 19679588930370230664, 1188520101254623950172,

    57200881656954836661276, 2272104309877844416200184,

    76508829174114535542941710, 2230380431623471505440499792,

    57247047715019314081805545598, 1311650968526319832649253222424,

    27135725398987488268270909185639, 511799816506329703619741885185936,

    8872553523579908741352970617029492, 142377525576284788813298353604925504,

    2127780996740784765407109426768902721,

    29772784890495778353150951377191105924,

    391882483702323928075825022036358168968,

    4872367474569576906815200432528275102936,

    57435298674329967116975193056973873457604,

    644037600811316759162265020598302344016788

                                                                        n
       Theorem Number , 25,  Let , c[1, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 13](17 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

230024, 2395446327876, 367312269767530584, 7951716665409767208643,

    51433341368775540114876264, 142981980855968931089278639420,

    210741100914371054431658423311856, 188401408248980837703585083133408275,

    112061131644894719071564606664991549326,

    47414081741699537063349837435040777248344,

    15005993500274268542353139651946623926275888,

    3693397793474208517216569258215913960816411071,

    729119673913304274846692336007998321649300969640,

    118368401673272562133414825379340530166490277893440,

    16131358800269735378714217188548969335923137354448792,

    1877434361116933079207493930963851220177105973430443643,

    189332222877098149241395389686638915953011322721005471072,

    16750983738736666759307841102868416627920068819488876417288,

    1314201666921006004885889962394049044090595449195686260196056,

    92285766713567721928023319903551290437355251831432363999541682,

    5847951839597255738698021989291487791822362895352044263157036096,

    336819648580836940404280034208500414882001683423486696769616493560,

    17745675784946406375346616131178182612225493585675353762527203480136,

    860137364916476006544051539694752660757143551313570846384256584239757

                                                                        n
       Theorem Number , 26,  Let , c[1, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[1, 13](19 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

671534, 21573981801568, 7250374869083769192, 297634382612778631910584,

    3350703464940927695247307522, 15292503814696763025798186546112,

    35443536567386905580044060543210138, 48185446384267423692599943414639958080,

    42423125768889059477734065073457537537992,

    25980893777983481607875215642456055030518174,

    11678536429181167619289908974305330422454918627,

    4016532351324313081473352801321870860018623061176,

    1092327243632688202767097286891879296661019004939842,

    241252252622747264847694582865690433674031473730769152,

    44231817287699119374719492887501318961167038852436818149,

    6856406114731503841581340760759675811576261255531299301536,

    912596530044023929251888489377390919686692526216694996551088,

    105687749037278298077814850891566348546009773048099189147834384,

    10771773890973182742861812269522717881350944678155367320043888919,

    975830814691710106664923008846326183612433785749494579048765151408,

    79261356936202789009809120189678675546503630915600736423663571237980,

    5816722461765735471396248052867445607568629824295204259237007425052112,

    388311939060717986963833863085300710857387279519583653434595689415870487,

    23725015308663812690031517811803213268238769257769503213808396382089105048

                                                                        n
       Theorem Number , 27,  Let , c[1, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[1, 14](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 22069, 13247828, 2753459525, 301373598907, 21137230100021, 1062084523490723,

    41043470158669097, 1280523596828922212, 33404059478257644460,

    747984735621658029661, 14672403517335075295960, 256231388186207517112103,

    4036075392643413072057799, 57963428348855358705825695,

    765808287427780560507168665, 9379057630551885035315096923,

    107175378534452471586786302927, 1149118891689993239929588428551,

    11616916844316003931048643722383, 111207422529135411794847674524481,

    1011898236267620717938931282482127, 8781318828174680213931571829350995,

    72896432827721468776874020979893371

                                                                        n
       Theorem Number , 28,  Let , c[1, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[1, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

181, 339398, 123258780, 19187757892, 1727535974454, 104921641818745,

    4713359974513338, 166379151021778972, 4814728033622632436,

    117828960226529991235, 2496935002255017375354, 46673196505977046584787,

    781013269104408738248189, 11841842638901571908584108,

    164320228772720323133112004, 2104356031035241370593462993,

    25049648168472285358608156201, 278867347234454240483616053944,

    2918845207265295072159307269855, 28857154688605300479779236725502,

    270579102571022776858188241807045, 2414901267743927919940473723781419,

    20580845340895960865922561826095813, 167970767505613574768901355727918451

                                                                        n
       Theorem Number , 29,  Let , c[1, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[1, 15](19 n + 18), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10135107387, 69755776268205472, 17969263794255674728646,

    780643206049764625342779360, 10548583952262428186100452921322,

    61253793404334083012077137563741504,

    185945162844741781100632726760782040481,

    335950542987167072122411047136421582001616,

    395744133342018444785063670920393704122998807,

    325039827600360018450597808461958254576352389952,

    195895578498162757363377864659787150345670888913229,

    90165346602050456708896837931775814455433091155038272,

    32724765998539559492403419979088107554976052763440692670,

    9613133725198931575188189321701790813443571191009397924480,

    2335550576428051577126740401208919757175800927847130500689830,

    477889682733640356983741330660399649543536802261086526534555808,

    83631795021872324364786202994912213779612434144040868184073085048,

    12684165833632202364090833429459192486038346396698949861282643380032,

    1686435787379239388984598177202860481215561992320134492938282209153098,

    198533151207855715854038756627348946562582753886724992527143864014542944,

    20876823468490317616786855899220162874230858015864064443917705645334633166,
    197622707054242524155057676987874023050188156884216666246407454549527337\

    4880, 1695709297095618899286397884168431952371359828287720548812999488789\

    42340553590, 132706595869898145878714002977554550824531062329311649509902\
    07963595582659124096

                                                                        n
       Theorem Number , 30,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

74, 156855240272, 236655756068278976, 25131321415235835281918,

    577153851350763463580026593, 4708917197302580063822226656223,

    17935479486877793034503460308923633, 37793330454922660442115582615311075191,

    49394085191356000399726583833692307147068,

    43435667041630833664548042890442299746178163,

    27301361031949633810786426595941888106558324131,

    12847866624769495319511490949938463455742492824567,

    4695138609743518719619315609970570138446727208061561,

    1372134385048949029981723612087661475607208259898202926,

    328492299913734003371226967491820040368848139324366607835,

    65723609837374300968872540683300630294001825591579676990492,

    11176397207161193294589882592188883039184200311280936138877753,

    1638695066148939184890256788975328185397675979088260378506230663,

    209734857619557011948851913214432309138389803235407761312800341875,

    23684827163789046402299405094394967959223306268888021445812933341995,

    2382121520253879653268985628578736319506151223300781980089947509651197,

    215145727236035990384534150505842169635042209725278529005645323055490053,

    17577196640583947926671090921409032500331927224351018344887348933352984263,
    130750067110259882758725885711512576403118173799472723369513125924542095\
    6676

                                                                        n
       Theorem Number , 31,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2431, 1033209276189, 1026597848405476800, 87550582979522066195909,

    1745553348637691318591628993, 12871802354357230637904762087353,

    45389094454320836103584974477329916, 89952433522353563623703388759813880959,

    111790249789554968887163507038709831063353,

    94231363250825871346826765068054918174177396,

    57121142806923268867272368614906096781997952578,

    26047622394985557051046066316302832681489079407998,

    9258785656519941258834099991158251758006307185001960,

    2640022841213724946493612245653715812940338338367059092,

    618224775731082273074272730239441316579428718221290047488,

    121249059780317411873014480474129490355188036869052225766497,

    20247868749484742861294005525806439283222088224535476286156447,

    2919903274394267529547427667220786036457300663090612017594099584,

    368056963855050065653360977774029137846952451306919538053002150183,

    40982213675299768274396188275078384866631778117600865025692199629415,

    4068316450568360525647012470052726059604096541939662764911864723838868,

    362996382884865944708850684740998718912577658145334083930119914124379786,

    29321584329458260075584982590322678420548233773082243959287540291935286760,
    215804429010823318485737490678475403954035344866614177409206101905686155\
    0809

                                                                        n
       Theorem Number , 32,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11648, 2582154084991, 2112180236984787195, 162203749562073323055043,

    3020168955199588003130101785, 21200371661244558430744244129770,

    71990533824396157652425255266121538,

    138439952552929982223205399466681578215,

    167839430205146199730155218399428207609767,

    138556522269693319532355689971894656156974450,

    82501431584926076797490823394073781934789235275,

    37040340478409974183272917891413445670538543489734,

    12987027562885719943215014015157470563415348021367630,

    3658214233070340759839484056644131008272281522664280003,

    847338807722891802950727720015216639759030732046956486396,

    164548602384124081111505714717689114410641620260738035128273,

    27232478127260273099858106659367994154359180442280044367688426,

    3894933852790590694984309561132912150178319209567520886555261122,

    487256316569741210756995552937245088147948039464154845558827160619,

    53876458096861152609386135313876552266033594793980033212136328031603,

    5313730093380523691726365646278043113588280877614649941758183699542642,

    471262295181994120488017369244645121572709846732463472286513105526969651,

    37852663752067469916791094239739439725136682026210504370018394125212819294,
    277123057791190070625421114184228350205092489207995140975544269441287490\
    5168

                                                                        n
       Theorem Number , 33,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

208212, 15368271769404, 8737035528024117403, 548886225464919624247438,

    8952017163189116491947357447, 57085570015541858545395378590013,

    180048123757761993562468189501342104,

    326360338940795207643583679556494907991,

    376837766995276241900956514175981394073718,

    298556251858499737647403735271190664003414585,

    171602903724162432675990975973066576591615536270,

    74710099323888937782164443601328396049464737166512,

    25494024111810117789242311002823867790186072195375717,

    7009923347344170018720663780154766346632722717868919672,

    1588863225255585549133823774353653606431018960486330580883,

    302556978807055538342192043591320678296249443772165835317490,

    49186597995797668316432236407656552522225533879993781887900996,

    6920878226341346170687263850353776961501408797395225388455060112,

    852878430602021285156653893022224203925629086410615584062332421557,

    93001967509878132285021204385274996426586645968709256207595961358470,

    9055055007499329913048366164370636388683264144624239006778577039903419,

    793483127894098259279204228098372814389534254506275728456386031213681016,

    63022750667512348422058131653456787993682114983206578730704547740038666447,
    456568416988799182057430654976338319146356705093036206701825956721145630\
    4860

                                                                        n
       Theorem Number , 34,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

796938, 36662206062629, 17574558426254854656, 1002764317040512206248607,

    15337881021812607311602633709, 93333529091332234672164188055138,

    283922421120351675571467110925335445,

    499919607707925008874774270752334194360,

    563554777182273364540813377247209268936505,

    437524660409998202830341986515637438595561744,

    247132567944314912895366187197253924663710390242,

    105969826652908274533198920235645096166702856219039,

    35679251766709755680487303478388924206025705969686460,

    9693924651481956509029163457559985561036136507497451935,

    2173743695834480856056396133421100275391879826251858038437,

    409927417229884945798058035115135167708402403606727725403969,

    66054258416947697295907537266318153861965210350268712133221559,

    9219187874488874789661113947884459896628152749249382442876614377,

    1127654493940661947884497460355865999038950128286637812950486573112,

    122118965004224082912032882943570581741824093696253635459835892457155,

    11814062395025575034667250596147584187952798035186330626847953848272552,

    1029090303169083470168776426446209465284669868878342707712812971000456558,

    81281237352777235912022461741376693240067521790125800098600926571105404984,
    585770791509243693594516841717167774322201116281179265710614314628931105\
    1739

                                                                        n
       Theorem Number , 35,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9985716, 200250310231337, 69635981744557266596, 3302720928016280898348692,

    44604610436033716452881742999, 247730003583604281737374211105887,

    702074980782351904881312805787194271,

    1167650342554281144293920040735089795662,

    1255539154442068598072955191768059279724270,

    936547918296649991952054469487002689276385623,

    511098830716462073959693881716130417008267336941,

    212668365256080296720749482067398519742004020272070,

    69728034488682945461296649479561960257931547896821523,

    18501633016533893380427654111590379802034600552527160713,

    4061381999963489090788157573879415926462319373437177644597,

    751276226867044721900837290864031033997935513314868846928134,

    118949581667431524045268930027873749571417172573761440999136587,

    16336620048093062672972670071826948188430190779388058649296779904,

    1968821394490801762363899281084018239430930510248073339858051166737,

    210308896484000782352795373137580076365235054958517788264182756741930,

    20088319952109754708225876803673012237716788380444923977797739469501867,

    1729197383294425178068442838884008567486176762048557505457418218297620022,

    135071532840201529547807921135340635517315686364562330871600856659318599887

    , 96335033501084287832582516559605005866142071015454286460659106846475274\
    49581

                                                                        n
       Theorem Number , 36,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

971470553, 5157923734070878, 1011252277056840331761, 34068275549089403209460225,

    363890467131555644034097289933, 1698042036172893485471393555508117,

    4200284355367188888229369573794391408,

    6257014036768425279095172412913050019044,

    6138902884503750525727107031345100047853076,

    4236416913701263534823523972664271441136507638,

    2161732860569497794991014924086920755575160513082,

    848157433080549195359354298033609915226968699377622,

    263996669080132930381441536277225507432607123739953039,

    66869990628353321931299059324457984311702780164693972019,

    14077777373207859235079192501311405548523529864862786508026,

    2507223255085878278408997340226767045581139163164053895342577,

    383472173181292405230655058755658188969437654202758985660171188,

    51021636919193291432207775888796368186414367001079672717859718112,

    5971724498521741070035308930102926399272774132805683834962556065609,

    620865405730660312995914078184578282104245090450511965398300453635284,

    57831246391162717404190088325601267630965363896830266324146729668514817,

    4862738500067694335636179724847419788370584675509516942628686854434573168,

    371600096140334177234048672964770747093237939869387800903123631722446368070

    , 25963410832227161752048022711483234905948963162593410802054959142827703\
    750599

                                                                        n
       Theorem Number , 37,  Let , c[1, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[1, 16](17 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2820182409, 11304319044687662, 1944367267117016697657,

    60449206754225283219332691, 610576610144003170721528531156,

    2732382845828764325674903420770539, 6540112542522730419529969470884380032,

    9485322369736977087526566196916848865174,

    9100726930023436840438046462144224340096857,

    6162051142805513519031393348429621095874531123,

    3093026890448738766555051833335925004452552113678,

    1196177319648280782924945784112675258103592198175878,

    367593894087539342435667035585747964617245286480216805,

    92053102344878362955856358616574650197429622898800583026,

    19180854804246655295507891286256123395369991373046287367854,

    3384285290878880393135029252455376164091708814688650945515931,

    513216881189499341050766687747929898852108509479098609879599122,

    67751504373516992356105789655529412539867965772137513745633661660,

    7872749211169095813896138763923832092685848305850770376478313393788,

    813051445752423721113643624388710734788565521250129776224684139914969,

    75262991304246115449287239324559259233676573950832429064438716848048720,

    6291868122443383077436867168361706958697622391458766512534508849206243636,

    478207249660712697217670721663345243461391541750765323634125009868725672608

    , 33242133179218817742719636515820785096191059507597751165955974580620121\
    484768

                                                                        n
       Theorem Number , 38,  Let , c[1, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'        i
                             '  |  |      1 + q
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[1, 17](23 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3728520081, 2288316744562763648, 11083328541094740654485248,

    4866104008777608846672830449824, 468440154520745836732585040226868112,

    15398630464977663227699778423475031070480,

    224313824618258617397513604555271120486324736,

    1714718520981499238330969264301381664817461465166,

    7733643528169549204647416029485054117264066556180516,

    22410806999050646893497359927874402815168856348711224512,

    44500691011155382560033462726487563752983555281463354845696,

    63656056363828624383410650031590561765592866856354036580488000,

    68260304688342956076567444196415208262960745820709465445313563908,

    56674140616399502068896893067816240756984774424972363915425316830176,

    37416767607331880921269616358794986731449539302598094878125199801891264,

    20086158223867714794690540120814891736338414531028272276675648720034315904,
    893464810842599140882419591472320335070486059631922896828716601935568713\

    5765, 3346768360279555684892314799895028900144607247478762742431118421703\

    007771875264, 10705397499772372544941819651470464403236098499916162161190\

    05991673859512975287808, 295997084802772238611272215988803230217438840746\

    980882030668796663716953251059696384, 71500709920556703898970949676027642\

    018857665766597361760960617118180529506822855865080, 15232142589657319269\

    422152092254917226878524299038060135062634864502293087650981085412000, 28\
    858406571001599620610278005509645613932511380711793236262414128174763924\

    70854080932237760, 489879676114456396243208994990409539182315036565752212\
    575269062095084104393399722408200385088

                                                                       n
       Theorem Number , 39,  Let , c[2, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[2, 1](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 80, 1728, 21776, 198928, 1456816, 9052144, 49463299, 243539760, 1099495408,

    4611205920, 18146576144, 67542995936, 239304936992, 811317542455,

    2643620028976, 8309597789920, 25275621036160, 74600683091504,

    214154123742704, 599168067670768, 1636812907973872, 4372982702846304,

    11442190721297984

                                                                       n
       Theorem Number , 40,  Let , c[2, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[2, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 56, 452, 2720, 13504, 58328, 226344, 806080, 2674280, 8356856, 24804804,

    70395520, 192026784, 505657496, 1289970400, 3197679488, 7722003180,

    18206079416, 41987387584, 94876233760, 210361233152, 458252167600,

    981913651464, 2071662657600

                                                                       n
       Theorem Number , 41,  Let , c[2, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 3](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

53, 8667, 446606, 12723004, 247840877, 3680250614, 44442051878, 455286522085,

    4075206180547, 32564312312577, 236127994179240, 1573581865533489,

    9735837318035119, 56387384315284512, 307805113693246482,

    1592705450223292382, 7849898855714632276, 37005267230334806700,

    167452613814257577264, 729637802739559376217, 3069744371441523161211,

    12500589514429507761971, 49377503456701611990713, 189555140765613692252637

                                                                       n
       Theorem Number , 42,  Let , c[2, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[2, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5, 890, 41249, 1027130, 17426114, 226002808, 2394616237, 21631469226,

    171556564012, 1220179848110, 7908115453184, 47285922047198, 263433133556844,

    1378313118305516, 6817328693994681, 32051697622971008, 143904106683559125,

    619446315441056086, 2565250602217442380, 10250526249942538900,

    39626712374705099659, 148544872803897562014, 541059651413185956605,

    1918427394972333458152

                                                                       n
       Theorem Number , 43,  Let , c[2, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[2, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

52, 4564, 157985, 3311126, 49938575, 593599658, 5875928665, 50236463150,

    380592541576, 2603738292912, 16318310960630, 94751118581898,

    514329240975911, 2629306086408580, 12735979718370195, 58754252650224948,

    259270036021305368, 1098485250591513908, 4483020054234156834,

    17672947773269222376, 67466534573044549657, 249956267046883806340,

    900504866755523979000, 3160205216592439424786

                                                                       n
       Theorem Number , 44,  Let , c[2, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[2, 5](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11, 420, 7614, 92353, 860121, 6613470, 43870299, 258494595, 1381019175,

    6791977522, 31107319677, 133886523807, 545477835236, 2116231200714,

    7856614574811, 28028128686781, 96420694775103, 320833446250626,

    1035292186467714, 3247259153092392, 9920165088726090, 29569617553154505,

    86136938080776405, 245566861459534737

                                                                       n
       Theorem Number , 45,  Let , c[2, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[2, 5](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

835, 3439323, 1852357506, 369191174280, 39398810120262, 2705499569652570,

    133135627902017765, 5033316920731354345, 153404286178406614407,

    3903306731967171020793, 85131803995258358551486, 1624452751426797795735147,

    27564433484627997856825321, 421455461651966888174798900,

    5870056071023200604392119306, 75157704936114802667015514930,

    891437672247896356822840789044, 9859534611652681745784944520915,

    102268213629485329249959064002237, 999757865322123688301934178809348,

    9251357447960451197343319258710630, 81346365134419487211293348757616658,

    681980924438847573349068500175346488, 5468003047788058417773479323839678981

                                                                       n
       Theorem Number , 46,  Let , c[2, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 6](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6, 16776, 4645535, 486416784, 28444304700, 1113563091296, 32331619411822,

    742747064968688, 14109094259561954, 228745805083129880, 3241035118848609675,

    40871556105207136256, 465412547578731909650, 4841717038137468866648,

    46458592811858797806092, 414483050367575509102672,

    3461385526315260843659254, 27214170873549977483210840,

    202439916797977262067272117, 1430952449908580632614097936,

    9647654978321723074083958440, 62248863680587947535730049024,

    385510897930402771446052598375, 2297637678785278536019564736560

                                                                       n
       Theorem Number , 47,  Let , c[2, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 6](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

91, 96272, 18925316, 1634115416, 84037430866, 2995609693664, 80901681678450,

    1753533871438952, 31746049694376952, 494188699897025264,

    6761482141757444930, 82705804582962884752, 916778609849720074872,

    9311245693427522381744, 87440085086188887150808, 765021700572097284433400,

    6276148651133182717545875, 48546977978590206662180864,

    355752819661640171372105768, 2480006447881866806753559112,

    16506542534464864282522052603, 105233594795954905467718351632,

    644448637330751830960664261716, 3800723224329990677126674874368

                                                                       n
       Theorem Number , 48,  Let , c[2, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

296, 219759, 37260960, 2948104990, 142833084936, 4871553879430, 127112061307328,

    2679460304481560, 47398998119399312, 723501791742657223,

    9732526375810515856, 117294920338582570368, 1283247865308326098976,

    12881493593049447675225, 119698475722950882952128,

    1037287226383952971149224, 8435876908031179187431944,

    64732848453530293750623476, 470878434664437204084253168,

    3260239864554852989137329012, 21562501701763025722163552256,

    136656370449273355983654434699, 832266060928481227677639950448,

    4883005634077144364549908846010

                                                                       n
       Theorem Number , 49,  Let , c[2, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[2, 7](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7943, 77040874, 90102376263, 35970062616331, 7229028343733669,

    891129442104530466, 75752889943990805867, 4793253129413307734479,

    238078454293973625677675, 9649720833478491390385541,

    328671633394895823341580830, 9624761505866779549917266204,

    246780687075331469608003128178, 5622845171743603158158065561530,

    115248608858124341766841647570631, 2146828356115225819813591814977577,

    36661504851387446511245935226073940, 578232673308569993589979043490023645,

    8477434001555989251183054318073521569,

    116178814672042089828370115997738079315,

    1495630587850121762237910169319689431745,

    18165285802170439979766585667749313500840,

    208956834011303654393799179636741356135622,

    2284362344206672074121203809485616161096589

                                                                       n
       Theorem Number , 50,  Let , c[2, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[2, 8](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

21, 1614, 51927, 1041018, 15271995, 178499382, 1750506882, 14908007352,

    112968252729, 775472590728, 4888824414864, 28611882247146, 156800184931026,

    810353955847566, 3972652326251070, 18565629629280300, 83059398469729212,

    357016734802439382, 1479011990875302090, 5921470747953486384,

    22967374668509508150, 86486161003790841048, 316784453094014381217,

    1130590306470751524186

                                                                       n
       Theorem Number , 51,  Let , c[2, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[2, 8](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9, 63064, 34445034, 6389146008, 619427544979, 38377161657846, 1702565282679030,

    58126180067130744, 1604033142046567104, 37065497573053217592,

    736411552808154177501, 12838866281964269794008, 199618740839469047720787,

    2804209754936103945844792, 35975860430804309022985943,

    425292120570428470981118808, 4667874050735493036266514625,

    47874835155888937752476597280, 461387897928095989287513962853,

    4198507221189077984954437408330, 36226983139203317786850785545971,

    297508717615750419276440408442960, 2333117054741987116147090551729796,

    17523736807066368794399240475051648

                                                                       n
       Theorem Number , 52,  Let , c[2, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[2, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

15, 16885, 3231133, 269692738, 13434250309, 465421456438, 12256523730461,

    259798520953193, 4611229028995035, 70528248185602526, 949872014058053443,

    11455501922471348510, 125374771043720967485, 1258806522899003864961,

    11698834664859433104397, 101393238266113781286419, 824724691360463541981129,

    6329905991046245714179538, 46058329648519914393646095,

    319017173966655111626008168, 2110912749912386267029872319,

    13386003305612603556494151461, 81578777714547241830829208339,

    479005280222801527591518189795

                                                                       n
       Theorem Number , 53,  Let , c[2, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 2
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[2, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

354, 157817, 20469966, 1355480300, 57599221391, 1773102888803, 42568922271642,

    837101829454730, 13957557419876577, 202425710966918719, 2603789458585924753,

    30162423163053564530, 318544304716078999471, 3097888915036322858634,

    27974560164845644223127, 236209380363976689149886,

    1876071156166025349881208, 14087642692711503544723595,

    100458711145168934528751949, 682932585625204171414596079,

    4441074585755661882608621227, 27709466112709279208171191772,

    166327793537000974465525214897, 962814740315255865107653604764

                                                                        n
       Theorem Number , 54,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 10](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1408, 4499030, 2455411200, 534133796864, 64105647843668, 5019881271091687,

    283563966857615360, 12342663297026842624, 433518192251505670400,

    12709723845639246015018, 319119101044552876795904,

    7001224882602798089679976, 136385105794790587976630272,

    2390048968181685453183191629, 38088234004249541126603866112,

    557006407589870612122340909056, 7532875351533601721826857232000,

    94833673909800415097174285693282, 1117762939472142053736701463019920,

    12396232200588986697157442094964736, 129923434207929648686550685693796352,

    1291897528408742827834430099559180410,

    12229554067511288481091609607218319360,

    110554425406026815387601523623708360704

                                                                        n
       Theorem Number , 55,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 10](11 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8, 7612160, 78376048979, 147341329228768, 103172044209855408,

    36979971568308207104, 8088006083588611100466, 1204382743489278047916032,

    131470900193530859659097848, 11089846214758461775799920512,

    751743673113903323566313441405, 42201798810285010063030530523136,

    2009131413148560433928879834817360, 82678549041409776436895890896052224,

    2987293848411029418354173693752042990,

    96009528220230565518726215784662163072,

    2775001126864597683286838000214308623584,

    72808353731810350598928138951530275400704,

    1748071446090593495696417860467415530795548,

    38674689084033382330511587188630169852979712,

    793294123256478165617238169656615067881922560,

    15167550232565915476486957701191469657663248488,

    271605778622301146128581597356597337224101367192,

    4574552139899023244787986846891110771225880023040

                                                                        n
       Theorem Number , 56,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 10](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

896, 120910588, 695090889472, 953534178898656, 543402857892177272,

    167886259131064296636, 32768015058432614930240, 4455324925292689693432712,

    451263105913962663562295808, 35738423373063159462499768205,

    2295014764919354973020170990592, 122913105774257025383487381302928,

    5613870247175693442161986896529920, 222647457492087657946962140213383698,

    7782443497507183578876478840957112320,

    242741147841934503245240791424508782776,

    6827362737257110816292117582707061738112,

    174716928612515543068671167456503963550125,

    4099634472421647873138474672383450077040640,

    88797843107884721203320956089854691884977008,

    1785932397541969449810302800632865023614850104,

    33526639790609796892626410977614096814514523482,

    590175275127054702923284840165428700544111300480,

    9781968738417996818846250832710765223349342533080

                                                                        n
       Theorem Number , 57,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 10](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11968, 679288376, 2811283827456, 3194469899785619, 1604042147124888576,

    451663010775562931920, 82041437727620283770648, 10527698149569995906662910,

    1016425803372277906096537600, 77298939024664277705602910360,

    4793647950975928368435272784000, 249027554930933284945666373091315,

    11072029839692884622687823527230336, 428709987973441154308077914137645872,

    14665389356483834547938042959078507576,

    448576887128252362527910525339836901170,

    12394092075163648005865355212630355414144,

    312040994599033667363180794557623751954432,

    7212693220015868533520440836543431642186240,

    154070829900093754117547242167382701347880178,

    3059003273661765693200800810480110418460727296,

    56739275464668226839133646982034675028659888128,

    987628830459136444032635185234572296163371283496,

    16198043489348185962681634823083838018755241454184

                                                                        n
       Theorem Number , 58,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 10](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

39423, 1562534400, 5565906148704, 5788582694301032, 2736273162259083150,

    736811985902741860352, 129256860439578289337504, 16126028101469702686295168,

    1520940252479254379298394673, 113394653785486484870928002048,

    6912760349983398354873951500256, 353779443770466242530363505499208,

    15522586954023260616220061450343398, 593975316149966995377047517058551040,

    20103843616204500909112827248376513520,

    609025168539120503986740574103603804928,

    16679929039540375062111434569251253887651,

    416571171881253065665967272094889597007712,

    9557586288982677832429317815687873053855552,

    202761303989860045728876837479628574040963240,

    4000096599902918453370035138147319867599572230,

    73754429287968264346333465500595428089686872064,

    1276672035331509128165424087671867900738478333952,

    20829529646195362641256558371821768837560318459904

                                                                        n
       Theorem Number , 59,  Let , c[2, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 10](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

122904, 3529643020, 10912071462912, 10422472199687792, 4646309778864144768,

    1197771022737512026682, 203075791573806376142848,

    24644617427846798779675584, 2271474769366581973979769600,

    166070043795844868454738427672, 9954294943167655217488031852800,

    501956933479180284881360501199528, 21737554837388563515458662257500672,

    822116608987381799531000972446751170,

    27533863455595079481585945748225706528,

    826175003689108803360386688513506915728,

    22430704544953950593850052819324985074392,

    555728438737623327209111743587513233597936,

    12656629083346446682738091532902093284710400,

    266679215315849851996826629080279453499994760,

    5227794082914266722871032474460808987571328768,

    95822148360608933547182751318307681971157540602,

    1649501552976774999557466823885870088466518949888,

    26772999349710412884824985738512261668376572958800

                                                                        n
       Theorem Number , 60,  Let , c[2, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[2, 11](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

34, 4511, 228731, 6854055, 144917214, 2375513521, 31977835990, 367320136773,

    3699361379555, 33328082504400, 272750265321861, 2052274469064352,

    14336263684787731, 93716662821284060, 577092909304599379,

    3366186575444608007, 18687530509897289781, 99141970696361136812,

    504419735079307615300, 2468880833192671027425, 11656451014562101006510,

    53215841467295122980707, 235429173270230266840912,

    1011261739804280436606020

                                                                        n
       Theorem Number , 61,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

302, 1169625922, 48651084358551, 311605247370761532, 655610709987131086729,

    642563623651431421529054, 357021002756121011890455005,

    127267891416917073415804691480, 31659613728183677470333084126729,

    5837586242410064608353847062199474, 834535241970659146496791773340084372,

    95762446245600052861457964690573845654,

    9065268860420850451796114308122012634246,

    723764804578337709204129029624875995568472,

    49627538437905213789733975590760055764769911,

    2967006950744324510317737372998623321977641002,

    156646682730057115604603209012769619594162930694,

    7383346241484785103180165474411010700513069719288,

    313604520525916794442579455861114850870623822183271,

    12101580335655919605553908910757499531949579197181908,

    427293014529557650551486349548500522123767839185745990,

    13891838068475561708459429125247870889703382302845800002,

    418178369406103554581539811346250274193624913426595105680,

    11713520458315068310223265675753277911458755822888882323890

                                                                        n
       Theorem Number , 62,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1400, 2930164391, 100610330210734, 580531342433108820, 1140669957538695565508,

    1064085620808473813369135, 569244811326329825797315374,

    196866667659287193423053409681, 47766889151453668332200499696114,

    8624366920536452567406664866627701, 1210898781812767762043508146913451296,

    136786005075954761514559098723526487575,

    12770898172693421979026288819214410914144,

    1007148415773480085228545883589458011568027,

    68300078111957342375780218739616516028506922,

    4042766550962442119296219622101461637599452732,

    211510921541528838142893941518659923492095338038,

    9886685800741843831378737461271395271119002110550,

    416730371427719096868126466208127686255054439750420,

    15967716052327692716526770401701498840519705484602737,

    560113830132177126674036082323957155619830344031941510,

    18099070789427486640313724962504249039695834890628823288,

    541724904654999611057962630972185665875096223628205607752,

    15093206261369016963171429037739145058535464371925864507505

                                                                        n
       Theorem Number , 63,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

22566, 17302133929, 417665598657568, 1978073651221642533,

    3408572245121295949864, 2889878646772365693855460,

    1436112611378180125874191656, 468138055217447423950932761337,

    108170821973118664835063569081502, 18740880047532539859392354347437561,

    2539615077168255113670313007538325464,

    278146421731891147440198102345732610650,

    25270154978457488389355078883584947279458,

    1945033334261748730296322949093699400603482,

    129054659001674463888395309191368997572705996,

    7489464458119399879139019995773597272803556131,

    384848846873835055826159259748063177921315523430,

    17695013759469831781963060937144942810883157999350,

    734628431773958548643780620499968438372587383610134,

    27756532558313556616122869956572789446807242295287643,

    961050026097503608822937732827307630090478293173275956,

    30680359435692043352272100464640265906437294754248911116,

    907950331424464905493147170658872989881013664091066917570,

    25029509805128917341052637973595570210145286213685245999586

                                                                        n
       Theorem Number , 64,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

275258, 95017149537, 1670947018200936, 6583991003276021347,

    10018216903325076166598, 7750692464145162108721732,

    3587331903052400908307238494, 1104220226839594083785991555243,

    243301453147381415775235265243508, 40488810518899211726267737766998692,

    5299620798675755161808982802227458738,

    563103397605881031834867748488019621262,

    49807292239776280581299840457635863193370,

    3743148084506874953073470682927945955643132,

    243081122318055622278509904235613551584469778,

    13834827402671254522914206300037739326832146706,

    698402979584136203949311967451404559530135867456,

    31593845766379569415083519474474638799542598159334,

    1292149055024154561417514627037043433002754522073472,

    48149397860607490545151733080671442389306366386932618,

    1645817046386971648526324837445595401195726771743136840,

    51914190944114798566651254456886610899570366280124263205,

    1519205525156177226250901546822374462758336530160804853000,

    41441828105614395242299119865184539192352996257314768069395

                                                                        n
       Theorem Number , 65,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2738458, 489501949724, 6460766293583474, 21436218001945469749,

    28982868391813711327598, 20538328787970082136202847,

    8875393466449495496775748786, 2584145680630651296599881908409,

    543631670650361666311678832786960, 86979966768778401850040525905341634,

    11004831156765307630852231900129032044,

    1135067718120544783753564748956761482004,

    97792622223417381360317821168563305141454,

    7178735350832064226687631940620275185324913,

    456430385128693454790844244198285800563297090,

    25483862352216373996041172837988569379709724641,

    1264141403856920098559283323189438295975292763688,

    56275410080630833026541649441426630724171378494228,

    2267782052232290634661864210933981106859418205646784,

    83354660465991585886773285634622981333767610026226442,

    2813144003566406861694416148232103919833741965838997088,

    87688213945240937166044462932219488310067971316746325875,

    2537752654810563504947944814047993452213120565389326017790,

    68509002095027600453874799930879324466723286613181825763572

                                                                        n
       Theorem Number , 66,  Let , c[2, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[2, 12](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8122753, 1087176998096, 12552621425356058, 38375062309467012266,

    49015938959398007512199, 33286993945388406188956542,

    13911517650495301994204462070, 3941801930843367525322858259474,

    810642047217621898610248977031903, 127220100925795919812171664973679862,

    15829398225460641602338020191927204327,

    1608955979742464770975260573522247780136,

    136834221949247388974973506435887877717516,

    9928846088196540507454933978106756231266824,

    624718978864309066535557010526164210140527031,

    34550520900041518818287554236577156351432759120,

    1699114796771796613221752926200250289615091963762,

    75039946402061803323287991222808030235267845169294,

    3001861364624499315365611261800122493979589594479213,

    109589774398307154901201260451344366928008834138944400,

    3675280571151874026900513725258041966428052411163636458,

    113889139921493749158070009175229774816270692209915373112,

    3277911688348229155794953948088180530610175173648243599278,

    88033874114820380278754035160416570603253384938401502594906

                                                                        n
       Theorem Number , 67,  Let , c[2, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 13](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

19, 796746, 1682145418, 983773702248, 264651323302565, 41558817847131951,

    4362290566164982820, 333602376670955875020, 19714427990153234372952,

    939318708002957909042180, 37250744130708670958368710,

    1260187331161393671842109755, 37083634823070314336741994058,

    964333084238608135145041756455, 22449477879216370942952939334535,

    472961360425233736338700659195955, 9100457147001950160834538261447120,

    161182579034320144399796164660118396, 2645591775354353759996554351942186629,

    40478896032574494084027501021605677315,

    580325816190229635031829491083405347669,

    7831158380944118802632586524119015136628,

    99871866639569521267126461221712881422342,

    1208057532654628163175673104016849867716584

                                                                        n
       Theorem Number , 68,  Let , c[2, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 13](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

707, 8534353, 11483792071, 5197227701425, 1177139981170364, 162941124183705628,

    15509855517875255974, 1096224200331860285609, 60686096757202958212505,

    2735855745996736555745291, 103446326634776359761077277,

    3356836371659428126943162573, 95212547213202524998647542978,

    2395940047073746943793601282039, 54152876974988029388685363293757,

    1110735913859314425039342583023170, 20856575028998876305115214456250961,

    361221700704767320060594181999694241, 5807900101119834339419034622698552618,

    87183462183094342008467792899051564738,

    1227934412388788818796942682196079370331,

    16298540672693124604709999836051535015862,

    204668084092692990332632189199966010412684,

    2440030007639116358389691881809378348638382

                                                                        n
       Theorem Number , 69,  Let , c[2, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3434, 26223315, 29032902619, 11695560031983, 2445369922736469,

    318946993328060894, 28978807024763287334, 1972299037913462986116,

    105803501577012828667789, 4643927807071269019897808,

    171579021256035638512420814, 5456090114583662321345308621,

    152003289447232110662107494938, 3764156886760323082456662019636,

    83856033707450931003060940447797, 1697560982438049560249643368472838,

    31495971283671669439639180520793172, 539523551732034274241022780161286525,

    8587198200489091986140616891943861619,

    127698950104049486031150254742807381749,

    1782936596059994880665612914448704392746,

    23473144515568764906337225020877273608171,

    292523916805355248918003843450854480604978,

    3462570037715789373639301056682219831839210

                                                                        n
       Theorem Number , 70,  Let , c[2, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[2, 13](19 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5533, 1914768100525, 1607159682271309460, 117054917451695193926304,

    2010138463267665065758797913, 12858608728352425097313963302520,

    39575627631681301484888337298074970, 68821868801054043233787784879697037832,

    75404030981518470122548701284198003836252,

    56270201437660976250672299763634650342326161,

    30311113674272662932304599523556002871175152106,

    12324500187814660970910231123820212807245784754857,

    3918266774825338776879615941696578531308033376930534,

    1002111807936612006680331326472382906831225510126283879,

    211035633500960613712029798833132795575602337970971401982,

    37310936935401443076533889205759171110519996936726908105103,

    5629352989826612519667786431698309437351685491822000535810933,

    734981132279640113720736675133167180127674801020018631574514208,

    84042694586950659016694320989726212162309522722201146475210229027,

    8504599520731007894167936325605862283243064721143373919309265789152,

    768592277593031187831824546328298112135878140021149820449455946775151,

    62533580938370243656146001337057258264798623959185831254147247363126779,

    4613139170579964641438466191639666422307221634870446773961138817827643466,

    310528172225317789848183607865455369783267320973668293452558756759073483741

                                                                        n
       Theorem Number , 71,  Let , c[2, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[2, 14](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

50, 10416, 773001, 32507904, 936743340, 20469335040, 360918897416,

    5352431713632, 68754151953324, 781833275059200, 8002894572870690,

    74714512151402496, 642953383797701446, 5144374142951457456,

    38547277123051385235, 272145280177963376640, 1819711677106974629214,

    11575377104903998672896, 70320200423619942931783, 409363723441710825494880,

    2290447177461345837402900, 12349903682874792598004736,

    64322899368205733164965438, 324299768086313210809479168

                                                                        n
       Theorem Number , 72,  Let , c[2, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[2, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30, 119200, 61981132, 12281601024, 1329851827122, 93945113501088,

    4801736743722414, 189855805468502432, 6082246547515185382,

    163287168106778025984, 3768264685207160704571, 76248452995053630988704,

    1374268306476807502441740, 22347483146308397500618752,

    331359339865598933210334371, 4519950194542136051398396320,

    57147350980094904667179714526, 674036065495028038348354564512,

    7457837737874915579921872576395, 77784616692779924785788413541792,

    768027878645032067422632176195096, 7206086594530312479105755241904128,

    64463917939409812996965159996275286, 551480261036089364197183495507130784

                                                                        n
       Theorem Number , 73,  Let , c[2, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[2, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1231, 1697184, 562046004, 84963751328, 7633000781911, 469099965035520,

    21493598626449510, 777551639877027840, 23128366273830831141,

    582886698400549375392, 12735390520042636481802, 245618234065026495296928,

    4242481758959319914331840, 66410515815506960567827872,

    951451906699521609717686142, 12579670337853044866338838944,

    154578429563209437175318089038, 1776076018011938531512198278144,

    19182013294923054064443047705756, 195635043955635438936269827554720,

    1891820921738906780099447525995059, 17408242474593043195632355274604544,

    152919284405386251181426461461750094, 1286021069196197065762806290667663360

                                                                        n
       Theorem Number , 74,  Let , c[2, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 15](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

24, 1805434, 5978900279, 5140267643038, 1950620189372094, 419517748074324411,

    58963827396194119681, 5929903761781807323055, 454032941835313813499971,

    27678616943375132738891667, 1389321606305343856262471816,

    58931229105803785870716893569, 2156353274020192520078235290929,

    69209701036412239197630636982543, 1975363976262932992213409696850439,

    50715205171609717490871815525579897, 1182631149069576610535474502852085057,

    25256805718881309956318543609726871847,

    497542602659259670950762036584046527144,

    9097177365730613884032641641103870726947,

    155230265285182778441317793572569715312505,

    2483878491728404808701881954634519431553164,

    37430467067599027910480502569985182814398590,

    533235461709160010498198824823918061331805386

                                                                        n
       Theorem Number , 75,  Let , c[2, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[2, 15](19 n + 14), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

258793142, 7622327238676884, 4123556157150507364621,

    295737776888308012835075613, 5854021239942531777835869298513,

    46382358870943411588645879901088887,

    183305169400779878771333381899385562632,

    417029796388536627312166947974171248506308,

    603403807637808173860556444036938841160092082,

    597113859624909932215764953830525996560832817405,

    426955712286159351444340912143681323747928466530132,

    230231220430800303390163670348221396519068260761787975,

    96873504841361458064590480679730530265421488613560675604,

    32698732042509784803720272502993138193939649441245359214390,

    9058810283271151148787966032287939191489841974733957449347158,

    2099610366387423751802880543830383276093470839343719356215256541,

    413787417446132715239375340728607739841338315243892905960496658864,

    70310146577883861982677883937151355885872023708488236881802529809688,

    10424921086475747339650282203644918589272855437815914799346432304393254,

    1362958187335236919863316929900193958759086762027510094031579918872749275,

    158574235773056641060484971679671409487568165566139246369764437969702375638

    , 16551747347766607871305392512259847562270866968290886832979289758202930\

    580013, 15611516284411068757238266209294323763929500292101299102282538147\

    31496504753554, 133915848548259437479552355392541052053502639531199051968\
    238613876159041975310255

                                                                        n
       Theorem Number , 76,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[2, 16](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

102408, 55512641055, 1373840698222728, 7288018022022061501,

    14529883271247976534494, 14445619323585857151740773,

    8464077089124494852537120968, 3258372087970363082827713915321,

    888820711331988125795702312643736, 181531716718416872279555220692017259,

    28941521685784059776102575407554599256,

    3720658676892843361666101285117776055325,

    395810320273545239044014336476995917918152,

    35584238469200486638143102035384641328755352,

    2750915410076597766905504173550404260683454936,

    185552262338590001841049687464659211339030631294,

    11055610651192627505675616077991906754170639988856,

    588053093763737375507448920308090501761924813713926,

    28179666526257105704155047661780533839639343113138552,

    1226307823638825766167570024639530332740390485141676902,

    48802762522605969289748015694164993029613309993405804512,

    1787123502010522572007412684237394119169332936468333913074,

    60550278512529348276432455162686221911135591796079981193464,

    1907488649516778516467150736086449890075424781359145604080574

                                                                        n
       Theorem Number , 77,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2982, 1802502738116, 2264454344754794322, 233353033189457673693962,

    5476755763551143728112254744, 46711707482488206288673498467819,

    188078182561746643890459000704532086,

    421378612475330683538717502989825090191,

    587298680371278032445081728465440716253060,

    551534523767795455719260920246176150520065792,

    370398925533715208555320776095534051379259734816,

    186230904131974780985496909839049129705805938404612,

    72682872940916440686866080406779817627694539614507576,

    22671431452275764274636051323216989998938610599594611362,

    5788759292803238675088584388268871943299575978403613893498,

    1234257361452033157714038711585349928872333682055917743816842,

    223479594872527374559153111441852431945318091623888289028253872,

    34858228487819260864992558027225931423635983934685414606235974962,

    4742055527997746943846735576115515515710969411588019417091088911702,

    568690043887430067483172537859644575334098868496758874476674884322248,

    60688301940077003441712415706903892383990969057141516103958446444679714,

    5810900324269266891059430753433379383407910800867798489528184582145416883,

    502889755576247495145461937621813611548630892510855364477100743376713908832

    , 39594060024217158996461882801444549515946213375228349889139560300051412\
    641725

                                                                        n
       Theorem Number , 78,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

66264, 11426089354859, 9743181282356419724, 813518595981439812657151,

    16636304283585632038619208544, 128466725609256333229325544369275,

    479303152352709065942498595013970618,

    1010427616637148001961590012085223710372,

    1339440998362904549783018245304147720077002,

    1205867575930543521425826165252461725650057547,

    781029988672172610569727708033965320893408089980,

    380504943023594772387172830787587971728304291226658,

    144438183800908391326335511250112354533707778393546794,

    43954056940434850219261221045458259516316741943223098443,

    10976791824648978973943274217732631804712264344314626470392,

    2293973212449488023522943394065781308236313848364590409697919,

    407846284032585629224846151599251495021255173517671279144751104,

    62562167588579060359422767362009501196187738711596850688493125119,

    8381125768686072238851648602516411988625776407118745229295428315104,

    990941021792627029824067118254059445668364530175208111087563798250216,

    104366062278820974434939668687979731872203608232066194934723024014646682,

    9871291994997457416410001819772486415273637767850806053498409052517272729,

    844559429430472523959525840034009547879788534256015260813374707064738747200

    , 65785145679932531999066952434401434576539878970713569265323353268508671\
    665192

                                                                        n
       Theorem Number , 79,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

276992, 28082490314038, 19976548463180901878, 1508075717725425529914838,

    28850012058991619513846840944, 212246585953003583136316857531506,

    762891282916158398376019495292115022,

    1560907963556444687976462950404676803928,

    2018758100739676665260569888703315133832109,

    1780004088873092672703561524062026286324588488,

    1132464909834349150319136381652549634870865506519,

    543190233315020301914108469507029959774422081397374,

    203380115703332563410466024985001766825049022817174345,

    61138279972195370340748105364105198058032492280888598122,

    15101449210574441459722285814577447477185602955464166196342,

    3124748301343664175559974552627877439970143960893398893976128,

    550545159553447883809313994722310991099869395884364565236393095,

    83754760554239105373651161504761647481732609605317885265085579366,

    11134955287763211101420804991699876428970394096191770353160547905899,

    1307293157880800748826912846669969597055691537801813201036860641164648,

    136786713405705569077502119797797769239989195394522244557208530530169194,

    12859173212473905257122116170374456386256891731433967863434994728135898886,
    109394939599663713767266477800779121219280291021561965590641023727681449\

    5895, 8475749016919456892782417155866759781056628229435690869708659565189\
    7924469144

                                                                        n
       Theorem Number , 80,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1086118, 67996419856001, 40655571653846239502, 2782678797992751791844481,

    49867666466560180076017088320, 349803768631450108689047747629994,

    1211920117034512316266641055514413322,

    2407485026224780998725124360958560976975,

    3038586418637979228024413194775302762194506,

    2624541986204714701166271729400225959441415497,

    1640432733587714891840646487129767955032008157158,

    774768941306512769475080641837040900908731123162761,

    286157833242870891526702231211647260411963814106148326,

    84983142299214041272050084722951253033291010839724040042,

    20763285406789514490726391819636755427217095375879471974622,

    4254029008144527935141870616846876596653809579579804578183850,

    742795044501386485418570783513092579295314447982782450941646708,

    112073996151108439564394631710509913705056914184874200493508562446,

    14787267377352380304790295604711614395287833437474938224339058512752,

    1723952302708290522144757512698255904957383205531155711454041046358757,

    179212269153223201815237160172707704789136012279658672507097282422627792,

    16745652280377807041201076981742952113899830609245934031806298231150357544,
    141652367217377394211117119944720957032966396789198558570396393477883481\

    3840, 1091682988461439119356158196088106336045299298157907252185707928704\
    88862390458

                                                                        n
       Theorem Number , 81,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48183518, 888955745908824, 328543397483724814249, 17019412656854187560028336,

    252693896731979221824283217712, 1543603355559515735873473553571408,

    4803611940205269287873936839807811092,

    8751513153140883767526870172691817526380,

    10281325648793678204134938917615671583905871,

    8357199505233018892262310371708207437111049798,

    4957387595463449273179408550750693765669340433528,

    2236822862859476493947355224984259468949361193381958,

    793493297648651966343721780649107862361429137444614466,

    227322156928818632941472561520785995480734151526663185982,

    53771153862693274650298558980318144591371591271138980321416,

    10698479457481490060865211435054799212473138763804915341547072,

    1818804838897763835233088253703033166896420281993909377024224928,

    267785200246996722418287301751712865034547143233748119091184949032,

    34544172792033081832980778373718167969946725552371670401753701115789,

    3944137139618535113391226714825794677551550093322826752042054314405550,

    402143560811231567497384274517555376036549676252803792468320984210422312,

    36904125878521388543758605907815776721789166970263208134888873728530371650,
    306949180585330976167494164670501375056880065545992980880121970607184934\

    5763, 2328444726982667537393469648303552450896871027433004585591157856643\
    00345360808

                                                                        n
       Theorem Number , 82,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

156867674, 2041842946082988, 650550415766605516914, 30858094181867156614621239,

    431372839488500094537498441628, 2520055102889947037606754108440963,

    7573909105530823159424281902260814066,

    13415318805659552408698668330075749338481,

    15395660732997869614642338997298492084992888,

    12268186427236895264625907782051909400230147820,

    7153630421359935444848928050465351429169237639848,

    3179751046934155113222423291275666182643099405301050,

    1113128008548063491191038016562118059361824675969337950,

    315138076197058578148188215542956669529089772668406275499,

    73752911763414116568825082499727349327726482258402102891020,

    14533003521680716467358382792070500034909955604989763622219606,

    2449017211596098080457921436072104191229866488421553393017025474,

    357669780600807284205689457006090066241839523425992886093125839170,

    45796900130296627079382633692744689987989928623564631659860821493926,

    5193020922422724116828365247378789183665304551839554226152405080844533,

    526100136066841319154550083742897063411237203681155629513063452952199582,

    47992100906843801123861633438383589860204583575864660818861642766475502321,
    396950838385668427431491670167913275864221032150120479373705350076202156\

    8478, 2995449371700829419992544308671221793976181487444548536715273196503\
    66026457984

                                                                        n
       Theorem Number , 83,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 14), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

493138215, 4635337905614548, 1279998461165186742237, 55717730589651133785816934,

    734202374387903413869790233045, 4104819172643400379288066065301674,

    11920174452342646945935486184318660024,

    20533819120481894442767960817845410905144,

    23025095480070903118798477787601826742135816,

    17990036679249975052652739440617898742367482474,

    10313195484053175471576077962590489338094932322055,

    4516442861735150810569185152707412864196895211236296,

    1560373291296749621072411468549768511867459493503767306,

    436590567277450575356797464118165974880208637569479350276,

    101099928047360921450118521447573479838084356295996986317961,

    19731217819536033121830105036857479200738589905187667841965178,

    3295966274759122253182637191217688858553711434264814249763816784,

    477507728023455943630079508357220401230675204942719028347747518624,

    60689690916890789259092123110367074913832350990320857268026578100851,

    6834691829895290369034490384387430852801033888402136622666801272132952,

    688015487742173399926074137835887860905316068875431277621619207387442815,

    62390363351373241589631565667531109086660689901619666756620789527957083240,
    513179366911371366525689610559926791621367408713319078933961442168999267\

    3823, 3852375481006268423971549364420845179584847254057660344914036283056\
    51964102720

                                                                        n
       Theorem Number , 84,  Let , c[2, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[2, 16](17 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4441381319, 23110667139643714, 4865176710165312234599,

    179470825155009513165957450, 2108358409863176112417932533443,

    10818002865862591036156748300181572, 29368182928559385240114251478832094700,

    47894704188316936457574268279836661091614,

    51308582732153893343645285349578893087724198,

    38560942553531602825662133855616552426985223008,

    21375622457192144694211074178635060665148043044250,

    9089482469005687251139920669870642322816294947459592,

    3059485057830813567095809671951741317317842164359306535,

    836317815781872466317337048033001188914823065432831603218,

    189638136672680515838318540884086570735096215267990931030925,

    36312112989688119630952473097571449701395444704388900606674304,

    5961073780193032190384010041133811700992071527347674213118279016,

    849939324889716796790546014067309345670363552525874576505276196736,

    106445813170681326262348201461428322816831637015919271033918075273812,

    11825294788229279155046410156292644484921992267390605757881221133476672,

    1175406023715636221586313728348111260640626053557714709892668310177914702,

    105335330607139137242475261313659104046670383653873774373428594041524206568

    , 85688556957415130370673072084301370961719446507497413815873664756418468\

    14350, 636612919095739245138648047028421136648689384146485837690448707341\
    919358919926

                                                                        n
       Theorem Number , 85,  Let , c[2, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[2, 17](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

69, 21147, 2181954, 123236964, 4654133904, 130827483969, 2923624731258,

    54281065225389, 863919989073477, 12063767959773666, 150457983887128080,

    1699684436415144285, 17589474843375458655, 168304424082944863488,

    1500543250735042797096, 12546981823895985285120, 98941758101267088045003,

    739345256796098930171091, 5257118627537805111675483,

    35699506549724685754233696, 232265576336284951365360240,

    1451953848261194735037807747, 8743192029952005509619576918,

    50830755157001328496263714057

                                                                        n
       Theorem Number , 86,  Let , c[2, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[2, 17](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9063, 184106273, 454938365188, 370251423888633, 146195749160212524,

    34255843845392153013, 5377277924526850836480, 612563885193584278983492,

    53565721475261500318886442, 3747082298550859504122675822,

    216379429849685493825920087370, 10571508060585473854496369232371,

    445665324144114702136863973051020, 16474382483877267122126708802319137,

    541153921731737179818421243216755459,

    15973566365990322629962929287497027476,

    427741450468638171345113022535932478676,

    10476214151315734094961918363577144708779,

    236342832015021838804506802598390164391388,

    4941716666842629921441577708104529367311941,

    96287472921248309330991088664568899263448253,

    1756736560014052190210657505867988661316987447,

    30140312667391354097257240302960358085924132636,

    488154544232569407476737155835612267857380784039

                                                                        n
       Theorem Number , 87,  Let , c[2, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 2
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[2, 17](23 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1900044783, 2294279327396798955, 16302610883775239227487178,

    9505194657430773448315233571364, 1153833386190759454257867555032640534,

    46308209639896905474271174447576357855692,

    805414082693750103071280931734885504379621827,

    7230519517678777457363119877261150489911905010677,

    37810676362224005929097729114329333651570160534155583,

    125742214829764741525575122657087241703651874400606399438,

    284124999942378182275355786345265685738919351517860091710544,

    459209439794781651268776521734572996018542989049682266440430679,

    552997804171270771003253568601363895751040546169816766139990295573,

    512896844110898939610855099144187998485805141024737860832302442604195,

    376519319419355849909658858326891460388684661116871450412736225905057687,

    223824728759881651315785350572813383148465831849296308569182729700959069191

    , 10984538900896337743230497058531171524669030294439102275102777741988366\

    7445064, 4524642853224985542273112866787697088921212827734877728756715614\

    8795427106078785, 1586758051470264728964594018252763409524074399296878091\

    6500933584844051710717891750, 4796828393384830343154937828913054569579760\

    597109141483304478691812633063210673481065, 12637012639555383671897021127\

    79663414535471911325011414185909426286378987769120651543604, 292925330529\
    420110346656113531587102156033623362824062960606289524125703026133412724\

    724697, 60255974866150588713807941946732365661154714733538996987238189999\

    628608416291299474616748440, 11083719456079512527148251788150905850388539\
    743607960279320065086281721505795731077719275771317

                                                                       n
       Theorem Number , 88,  Let , c[3, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[3, 1](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9, 704, 18568, 289784, 3258816, 29103096, 218435272, 1429144412, 8358887944,

    44512265488, 218800366200, 1003437761520, 4330026897784, 17702454168136,

    68957012312163, 257146529591424, 921677043849752, 3186099074884088,

    10653815275522744, 34548799768974704, 108898426357335312,

    334299005385789692, 1001245515901618888, 2930373893017359760

                                                                       n
       Theorem Number , 89,  Let , c[3, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



             Then , 1/11 c[3, 1](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

263, 40296, 2073792, 59696424, 1177964224, 17719081880, 216588121712,

    2243784078160, 20289564025136, 163638700649088, 1196574355716336,

    8035101941871684, 50058729904958800, 291754500231938048,

    1601739894668762816, 8331196971005130664, 41256083572655415408,

    195323387520262845872, 887320564690091759824, 3880069547492081381248,

    16377080952227751588848, 66886254842638384286128, 264903595593298081319757,

    1019379844508386758434464

                                                                       n
       Theorem Number , 90,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 140, 4372, 73984, 871508, 8013329, 61316716, 406485812, 2399177940,

    12856216320, 63463329809, 291850266700, 1261446696276, 5161173486092,

    20106730572544, 74948891709490, 268414159732012, 926797528603136,

    3094673295458304, 10019224226075060, 31523863855554690, 96584883423793420,

    288683540801398528, 843087085394086452

                                                                       n
       Theorem Number , 91,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10, 614, 14314, 205574, 2171328, 18439526, 132703680, 837475072, 4745973834,

    24572777830, 117770297344, 527821585088, 2230113672704, 8941555875840,

    34206187974186, 125422882748154, 442489318949194, 1507007822099110,

    4968817921255680, 15899896337578688, 49486409266230454, 150094402114318938,

    444399268291967254, 1286401977394974266

                                                                       n
       Theorem Number , 92,  Let , c[3, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

26, 1216, 25158, 336374, 3382362, 27692800, 193679360, 1194286326, 6638737216,

    33814083072, 159783787142, 707301012032, 2955851995572, 11735857809194,

    44501556814790, 161872354235456, 566928415732826, 1917928490490026,

    6284764197896640, 19996240549466102, 61906131963703552, 186836414191630870,

    550628198034620416, 1586990341679459402

                                                                       n
       Theorem Number , 93,  Let , c[3, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

77, 8790, 371671, 9217250, 161204463, 2190502923, 24521279954, 235067037219,

    1982939697427, 15017185817487, 103668554932145, 660184155060416,

    3915519302646084, 21797251300200923, 114631523776529354, 572597171002150228,

    2729204126427574748, 12461780738047598825, 54697459087358251262,

    231469660929394773683, 946893431627533360299, 3753160700275604805628,

    14443757145395773208973, 54069647433660876878661

                                                                       n
       Theorem Number , 94,  Let , c[3, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[3, 4](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 2702, 522436, 37944024, 1564644945, 43995483415, 933143004009,

    15896681382350, 226907977961480, 2796799501890060, 30441018165688933,

    297646168788670466, 2650071909959935017, 21719824951603324579,

    165335472092424838345, 1177617442205129547562, 7897338870198265593080,

    50130711653915186595228, 302593100473362568737295,

    1743689179070487308781105, 9625882937715767323982956,

    51062269953714120594696471, 260989562413801642004925023,

    1288418867109277624637973481

                                                                       n
       Theorem Number , 95,  Let , c[3, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[3, 4](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

17, 14062, 1914885, 115146045, 4197893620, 108019204739, 2140659841764,

    34543301448035, 471560110757440, 5598084112747016, 59000244947931573,

    560955738134659472, 4872724493521987199, 39069613633827322330,

    291605718659419662176, 2040338546418163567866, 13463047639659055063703,

    84203156076867063866550, 501374460912675843428290,

    2852999584906665036808700, 15566775651844862017315103,

    81683402603364935395276435, 413279526101276892989346995,

    2020893136442872778936882775

                                                                       n
       Theorem Number , 96,  Let , c[3, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[3, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

55, 30498, 3576963, 197546092, 6802988337, 167905026585, 3221846873880,

    50659320320477, 676886033751431, 7891137155597983, 81879659840378110,

    767957367673542881, 6591138076338428937, 52284715043427336789,

    386499902784792185601, 2680829071625432004020, 17549328769718669657566,

    108964498673333534638677, 644479562063833866976166,

    3644668286944929987644168, 19772294145067788374348635,

    103196498368600348087265682, 519516963607543560926958098,

    2528482271490967066875309559

                                                                       n
       Theorem Number , 97,  Let , c[3, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[3, 4](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

161, 64293, 6583750, 335728788, 10950995385, 259665876866, 4829568240910,

    74048967808675, 968924196976223, 11097125931545967, 113397917571644856,

    1049447695462659479, 8901271955748284875, 69869242988533389408,

    511610102604506208416, 3518226490250081465256, 22851221181603079724064,

    140867711604510118206900, 827673341268397857210822,

    4652069780396261160197563, 25094180380388601153917539,

    130279607195312580711895247, 652613815632674840461711975,

    3161516352382781849782327901

                                                                       n
       Theorem Number , 98,  Let , c[3, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 7](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 3026, 525876, 37488273, 1570163172, 45607865126, 1008652301332,

    18016199445278, 270546296534626, 3515605751758614, 40392665082951204,

    417227243716718895, 3925780489094929596, 34007818465852670222,

    273605183025448967108, 2059329846181744307470, 14589984615890340358076,

    97811911584167959973062, 623307897481891380183532,

    3790501416252619797523551, 22073493451430621544789358,

    123465706849640437885911450, 665111203427502847411091088,

    3459078490986158694114129363

                                                                       n
       Theorem Number , 99,  Let , c[3, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 7](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

60, 27598, 3138636, 175943572, 6273795284, 161977759213, 3269859760014,

    54268503401550, 766834445231912, 9464365072573377, 104023091552634582,

    1033656735163721502, 9398676652473234724, 78969664252000927103,

    618128617400911898508, 4538106155984006195582, 31430565099389603802936,

    206375572435444482779026, 1290179359789526212662378,

    7708097612167857122123812, 44154237779175178226352248,

    243211233917972882557996918, 1291515974264478905220007678,

    6627066075648546888994856150

                                                                        n
       Theorem Number , 100,  Let , c[3, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 7](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

247, 76874, 7349162, 370887904, 12298081835, 300743512822, 5818039641378,

    93274906241048, 1280499140915760, 15420366829479350, 165919877958019788,

    1618267533741240624, 14473283959522861063, 119824111193617195336,

    925503122636594812038, 6713086705350991805104, 45983698118011032824364,

    298886867749755685556468, 1851130131281281521620314,

    10964063658855269826488728, 62301441164305683620144568,

    340600174301811185559313542, 1795996658691106449557447334,

    9155013258508190415274065112

                                                                        n
       Theorem Number , 101,  Let , c[3, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[3, 7](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

202260, 4730031475, 12066340576576, 9619965801454132, 3614875635566628740,

    792049940839132425950, 114956487596059879060180, 12016921619793324182360608,

    959290059963815749769566716, 61040112373220415625692929685,

    3198230259930000653072238888530, 141531080874745489181494898318660,

    5398074963659756444958940902069576, 180397004178487641347623451018677782,

    5354780943836666020897534132904401220,

    142802937945266828067754581508243759780,

    3454804777160262128909038810812096590860,

    76455185880023860114945008174374380152408,

    1558856552365107844229219493505418885349184,

    29467288257327974806255106223826387400291480,

    519274278813496722594303878317828624422188200,

    8572088896150338452854277195711231149372041755,

    133133178197059789075310147271052491617151840004,

    1952869084616244414173351588234020889597907606118

                                                                        n
       Theorem Number , 102,  Let , c[3, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/19 c[3, 7](19 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9865335, 2703928834792, 43663125778426260, 162665425631818920660,

    236206841345491659126611, 175029264059869170371339760,

    77713616354205124846950438880, 22964967070983708176622992566260,

    4859753661551711530806878347281310, 776911097511627072647135467840680456,

    97707273330972040256884713663535204060,

    9976657382609626400103498609140363744136,

    848149029443270936779660847126829944639565,

    61272890635895625851519880563393124778929200,

    3825676584598401267974809166920445237954227452,

    209379431022605992174283659291218590881268803870,

    10166021273968275052554282239518527403979534742100,

    442401757752671700049209929343635318172771650826920,

    17409405313248498269946258923548628888017773400879840,

    624322439951281828243799228940468147044405780941429832,

    20541772443634842359422576956271525290426533600558829535,

    623840384257904828307851225572087561455241389445408928512,

    17580392773083558168320513073323665271915658313755053640040,

    461923117078411202124841779974765777286024049924107999087720

                                                                        n
       Theorem Number , 103,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[3, 8](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

242, 750769, 337931121, 59038882837, 5677351858453, 357722969636911,

    16360455206817371, 580376828455517116, 16721240935630083644,

    404585776850573340016, 8431761009688081320101, 154358991813971319794716,

    2521459792624227177814806, 37222064753625139176854857,

    501804948859488368575470242, 6232618067750045272170582548,

    71852072209338317982349236044, 773764026569614472714031751309,

    7826534528367988369589439156666, 74714628595027810779239345676906,

    675998165394198393289307963157355, 5818414106966979729282409175087822,

    47798991896508808639032022357669326, 375898668094968365050083536058707395

                                                                        n
       Theorem Number , 104,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[3, 8](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

35, 7632015, 37570198169, 40294218937676, 17584546296885147,

    4151933953593899885, 621942808565334181520, 65318641517241722131606,

    5146598292455040644872241, 319330422001492598184390154,

    16176225467493977647922661598, 687850255453208388888076678017,

    25096354284865163683397337568094, 799649911322291519235910582418625,

    22576379102196916504098293609069796, 571626001048230068712650419094126198,

    13112616898907263477354920413456424699,

    274885371169074269921734800167874543857,

    5305667624835715808564094879148802919140,

    94900622406506740922010325922993694365103,

    1581981177773493167699373602762287724448658,

    24700538635635525428793471318248084500162025,

    362833830247056582000481081059719405758615932,

    5034079100784436631376676631535026055109976188

                                                                        n
       Theorem Number , 105,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[3, 8](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

154, 18184639, 74428403998, 72261063172192, 29552223262032759,

    6658882675768905845, 962401685064155580358, 98228449336163567142092,

    7559882394491866468560841, 459879266442885937998414745,

    22904235807599350153550136434, 959678072381070076852995179213,

    34562159985265454721198251965886, 1088609337505795420218192502149246,

    30417540297778332770293105136698662, 762977849968904122658387541140649806,

    17353476843384894898177151687432767129,

    360961259029888127938669686513644639117,

    6917243672807710765757234430620164421300,

    122908973915455888239256870005853654858389,

    2036319447999897224448300302787540167038421,

    31612983963914393823186169493118304363028544,

    461897180208084215431726181129435317435866685,

    6376504559674259873804675296227795897839959852

                                                                        n
       Theorem Number , 106,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[3, 8](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

600, 42315748, 145787017383, 128687857321849, 49424316459270461,

    10640789686310053228, 1484986147808340615203, 147376914445296868011162,

    11083298426906482163025439, 661198523274223333761324179,

    32384390361739033623700605666, 1337257952720124427086724342911,

    47545672742290425041334129067122, 1480522467180324032238042158963013,

    40945589814794531481903607350534452, 1017562205970213062700259156205708126,

    22948975987491693443713698982983023616,

    473669710051964409036545444423903341248,

    9012703596176627626437316163088726894656,

    159091574148820723720816106511837619646394,

    2619734381720417942076490398787607029329852,

    40439615164316165366038547506774657272597858,

    587731965060390439212763682720444358938835020,

    8073360303596390802084851903551434392079220236

                                                                        n
       Theorem Number , 107,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[3, 8](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7002, 215047077, 541874660125, 400084190464257, 136317992938188437,

    26886226025640613771, 3506166882033357530462, 329502810416142802861848,

    23686768479579634021634909, 1360193094903320493686404830,

    64468880489366671924652242967, 2586978647341328409167859353610,

    89683168964951548614722933851972, 2730422728575979362804366372525733,

    73999178669711752296168111986950005, 1805585986592009261293746902102603562,

    40046649372111458668370207791478014769,

    814015475266885926439851068267893983160,

    15271924511020676554958772245968213548770,

    266089842594634103990947980157295143641939,

    4328986452768728634421462324816705445294912,

    66075851937463110213771781593206165729722243,

    950257884940783012045606413268881873155657594,

    12924954330457790406035283346548481914538591077

                                                                        n
       Theorem Number , 108,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[3, 8](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

21639, 471120854, 1029278406533, 698766981981541, 224867542382090523,

    42518992802509621024, 5365666097074233567337, 491054844933555989212679,

    34531336181328653502691797, 1946257021673033229641646821,

    90773596233947388960340097526, 3591640648578997708803241379138,

    122973232165784334369555953743456, 3702627158978987100452442202328614,

    99350598230943517068281948157611365, 2402327325659995278178396640338671623,

    52844192336995190775017735400631915076,

    1066053591666669840378933281415949987649,

    19861600208958639160521164078186653236357,

    343834279496471800241157350487539642179539,

    5560409854580263867077063722628896734952415,

    84399455182901068227754497996872308477833442,

    1207457944437650180243639657168277823347255118,

    16343089651427783484432187433890804999685650509

                                                                        n
       Theorem Number , 109,  Let , c[3, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



             Then , 1/11 c[3, 8](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

177587, 2149390423, 3612860273561, 2093635817258755, 604044334549786019,

    105294044883341361322, 12467786224995518870690, 1083587057007758622206399,

    72990524173659973350078711, 3966211497659119718654007076,

    179233413605576769068354080371, 6898338689145986422410255254090,

    230479928308147374497962453862697, 6789461832308853171877630316417766,

    178624830444993438439087745837935300, 4242728903465532689585196640043845661,

    91818670767981494087444471121002440026,

    1824812159795636326890918920445462383296,

    33532563681455964863122970347133710633048,

    573138760669664773332841910267110106613212,

    9159408609251368087445707179245292349208401,

    137498470287979484635984161995174921655115669,

    1946873730132871423934320751315534998253557646,

    26096682368562918879482423492381639502123431229

                                                                        n
       Theorem Number , 110,  Let , c[3, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[3, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

453, 242088, 36116685, 2692140708, 127005092260, 4297828022952, 112579135280130,

    2401077326162960, 43209682093257972, 673603640736159120,

    9280980030325375622, 114811914243773438376, 1291419332240463658635,

    13344864918493251184964, 127774492810514824409850,

    1141779266715715484114808, 9580454307727100627302412,

    75882163438075305397187280, 569931395113470642105348003,

    4075345125081453614088214880, 27841146092125292039082921786,

    182279775377059105583727152016, 1146882401197931976280939225925,

    6951892122941305982507789794020

                                                                        n
       Theorem Number , 111,  Let , c[3, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[3, 9](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

272916, 23280087336, 150589754471733, 259078323360061200, 190683613203614202954,

    76560835153240187259552, 19378525990835489188917141,

    3398594592053522550498739368, 441108615845358452415745856125,

    44458406233658871154014260211948, 3608680702287503968048057577184735,

    242694565247059331866809790118843272,

    13833195011229848401224996622082500578,

    680659918714370843764904407310374598160,

    29355396010660204401274161394380567065657,

    1123902081989730881487412247771905977395400,

    38613744815894435799076407193148430692816949,

    1201560965690109309610328524211334054910289224,

    34136045683837461756770122450540349083031180490,

    891602024941307457005054334634790749825819336656,

    21541535678306173310428462315996898942120516219541,

    484036646040462869230692696727240552315079531580192,

    10163963376444491398549880727003703342325699314577445,

    200307211538030285892309027052976535957456510926158640

                                                                        n
       Theorem Number , 112,  Let , c[3, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 3
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



              Then , 1/19 c[3, 9](19 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

492, 33422392700, 6457591662832668, 130467735349841296758,

    708703852160321956201360, 1580612521206098818811209599,

    1829445295510869995396786787900, 1271679603645869599423451433985295,

    585821358663525769467899736286634256,

    191848881613702316142664326072865248543,

    47054597226943279165258837222240953482232,

    8996381351694891225478944243638110455240825,

    1383690014769551821353804884285346344148412020,

    175598558699549638245153981052028265691865782798,

    18772674921497520624057242453391877895842614611532,

    1720067091434638952408254324864154582870813129920827,

    137052302631283879821053544825720787760364981160003280,

    9614359201794547056953654739446732464199647551386447712,

    600159399628594852868550614283092756484239497044120234576,

    33645907122296001586963845982179758250086991902855609561348,

    1707734564359314307266400179538940009971106405735592223652748,

    79034167215778575843625324096981846864550656427840617836922740,

    3356198465225899489641291660124420684630714050558405440670131704,

    131508065608758599288081867075398964127162079531847978918044650081

                                                                        n
      Theorem Number , 113,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 28389360, 1312193284342, 7442726558272552, 12951940140799175535,

    10238771296292886053697, 4547062889282968572109012,

    1292907622370025835585334963, 256871355085361693701667185394,

    37937206099395699497103068849774, 4360186814278056034151646146144093,

    403872912628984397589341221229841340,

    30990605152471214576186464576476700101,

    2013955897837025615087039752017887381951,

    112860578337852301939984593787343590323276,

    5536211593923365935941412021800393774932599,

    240731349838372963082722589890272945464394602,

    9379002694261263971521337858902379629549108308,

    330433403060761365734538259078952977652310665284,

    10611607440608241265091001945530777344867474011000,

    312807433859506719237734985990982994283642855110071,

    8516013387443109853099494110498022812753560641832466,

    215287267496818366056675271832104061444552454353325099,

    5078367031161408273820095446434065727547866721042798916

                                                                        n
      Theorem Number , 114,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

46, 183134880, 5496762626453, 25003985189788651, 37807303597900245450,

    27048122103584148440189, 11136073343310894001728375,

    2981860191196544656711538405, 564002182454615160534799815219,

    79931632156116844730077201604556, 8868453866566933781695415286730664,

    796709592003347692815593306183219409,

    59512552431059405073970327988160563407,

    3776236936696285611136470550682237163303,

    207138717727460953509730853507954639926059,

    9966537065932342178244932478975413907576525,

    425835151871984215568262179584874768701807951,

    16326655467079493349315884624248586506764373496,

    566789205070465633171139512260711379435420668542,

    17955944912429867644645744439311660721643276414917,

    522667674504791205841486731539401341252688948530225,

    14063323165742300212249445066590518320117889398206198,

    351651127156572565222860806198167595376967674549358243,

    8210368043752357980563639821259310122382126487440220944

                                                                        n
      Theorem Number , 115,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

993, 1075524556, 22073132479353, 81894962034985719, 108452904303848539124,

    70534851177431120172938, 26998677711810457112839484,

    6821078322970253085355886104, 1229960483684261305563324296150,

    167441899535926925035092588590449, 17948439940127305713200330700490209,

    1564814046002403218259705952199099455,

    113844793453901843200368406297840029384,

    7056249744331039408162031650099100549902,

    378996325077225707952076002443290070194275,

    17891898801110966064225766922316366163696114,

    751342871017178775055368161187713845226420660,

    28354234212832192004752000267468442889542265548,

    970106636226146839648075763015757306906377311763,

    30322552143833383686568725134734483847372511857397,

    871698668883375209144155902883425812455703299689773,

    23183899884127138583595715405175893969420427223932130,

    573457712050333888758722935198406367755106195405939941,

    13253805815416130638922827128653942876280783630344755963

                                                                        n
      Theorem Number , 116,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48391, 13211786474, 165394604692258, 464448523358737126, 510900256728688900525,

    290235096167150349034830, 100080557616320354143828446,

    23252306435471303244281663041, 3912320339721257741611834568605,

    502351606543279169704669995958308, 51207936580738534817602178131681560,

    4273050350111656777515211524713355038,

    299086346953259001031416019752853433788,

    17910082897089588700572327568787339351089,

    932649318780680186247922643897260019713076,

    42813089541118384889000676002345810616019415,

    1752591605323397685209076601991837505228800673,

    64612300050842894563784969299335774792633347225,

    2163611177202308869288872193707993161387008836017,

    66296915431456168654335152605688848354976358172103,

    1871035909333132523979948421326571305683814776495496,

    48914705440109938446453327638466111021273060397782618,

    1190639545144086408592284248441170987131827876722779622,

    27107085147016514927407089396198370495202314527201116345

                                                                        n
      Theorem Number , 117,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

155495, 29461001195, 318044921363729, 819129182444141560, 849758083923815114399,

    462333470617866989382283, 154162888915640000293162850,

    34860783604800317142922169142, 5735257352698923928073217256610,

    722543283398966341975594049580441, 72455963508702567633579258583046105,

    5960130645354427071787961672795513007,

    411926037350279245602910541570599667432,

    24390341951214889837989839558270858628934,

    1257271274851013250909905588923377183866117,

    57186347495980655168580806765786935638859967,

    2321426590514731442357459106318527317614160587,

    84928150466947088877740886788634706871620282439,

    2823848979141217436499935714101404670203111609044,

    85963040402022764546717671417028932455140819425671,

    2411346407267199317462549017036391550179093784460642,

    62683849144992203287570488180107686656058834148834724,

    1517732889513234512641405976920981552691914857308861940,

    34382706789833772722460115933363349728705008306565648730

                                                                        n
      Theorem Number , 118,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

475763, 64695658495, 606612601295355, 1437069054683849649,

    1408046438666154531847, 734367493559842553892388,

    236929005587172863436904955, 52166869264807936269188608964,

    8394371376914192471296648748755, 1037851786258253007867238249217711,

    102400772405773469098243793726692379, 8304759293696389523312108621456051374,

    566820030906327515984093883717216783441,

    33188115635407042918402787737321049311813,

    1693629690178242663639552301471978724819919,

    76333676966591041488048998062388082481884991,

    3072997513020928680696944386750488079942522650,

    111568798982468434647697396175965134740770892712,

    3683639868440837316475604454548656038697958701944,

    111408975223165341946517105282302857835375212335110,

    3106287490582216063832060242555805097499989333514104,

    80295166526673368430984763725934998710844505069792165,

    1933924475395324956385859897466915619346947041690910572,

    43595015413321789874938250098001205065210959182106515772

                                                                        n
      Theorem Number , 119,  Let , c[3, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[3, 10](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1394536, 140032952089, 1147955001389857, 2508292185933039316,

    2324545894436719792880, 1163182071166883430748114,

    363312933476898248637215869, 77920258016570565766175611349,

    12267293822115982487834759904878, 1488774049772195352371612742742329,

    144553885258518275285674072995901264,

    11559951571416480404271350763666822425,

    779251892940516746429330281771461511137,

    45122647959083780565267057283701749341874,

    2279760313137365961416927918303158535661689,

    101823987466841154661447096540998552967388132,

    4065409171082585079487082274211479506279876448,

    146483941096690471332143569560449689562098760699,

    4802724308774602356698178118882601672762596108969,

    144317748622588240784380149342129993404927883602141,

    3999717375107977835970482155102392665143028432903851,

    102811482006108833992210419760379271258461958862972313,

    2463279179457196434899027366734687900137598717997841649,

    55255293735094827698333003812374419486777889283434534220

                                                                        n
      Theorem Number , 120,  Let , c[3, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[3, 11](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 139267, 274704640, 138901388080, 31655942729480, 4190728979705920,

    371053859333052416, 24001299712921627236, 1204008798946655205016,

    48885136682442527049786, 1658454964326694767436368,

    48178123323825029869368096, 1221841802310146274085061588,

    27477102636573916800759350496, 554975928359534556209799000912,

    10175411582631790144594481697548, 170886476378291324017244702116678,

    2648929759155475220392390996464784, 38151138682672274926518215179368448,

    513463617048919376842919254014982960, 6490188756549669411061754670409573456,

    77387752930974274907949994010574041120,

    873894002231853784622184113758707215616,

    9378576426725748889255888811482195668760

                                                                        n
      Theorem Number , 121,  Let , c[3, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[3, 11](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

104, 1486916, 1806020058, 700603580705, 134119666231296, 15648009745413312,

    1257215793435620928, 75232638976598194976, 3539237663927004663808,

    136118213889042935215668, 4407760113878396252274844,

    122952076697158054166035094, 3008518199134082275841320000,

    65532604986149562533393342656, 1286222695157057934088636607684,

    22979005058916306164545671400000, 376901973056665609655752600483232,

    5717350671631930736196527724010840, 80720238429550390993348844253369720,

    1066564911709807250747855904394278203,

    13252906066816474603386394267189588480,

    155528423107542009322979791043871959056,

    1730341626372202147673665465564289992624,

    18312595880336390861205066856800080358976

                                                                        n
      Theorem Number , 122,  Let , c[3, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[3, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

544, 4532608, 4474217276, 1539888562468, 271904352200611, 29892852800632752,

    2293314116009861648, 132214714645037336316, 6030592218089728420160,

    225938166688853132517448, 7152929756485091833669200,

    195627664431195797171687822, 4704023248088083790543716584,

    100881300691992578686089654272, 1952462043438381259651978783152,

    34441405084039896689103121480664, 558401800227185038786476483392160,

    8381074287406166754160641997176832, 117175344497854215322637133875237940,

    1534288572764384592850709237897340400,

    18904970907746232374124363031512279583,

    220123902296371607028500201394449346816,

    2431097216066523206180154239019147845056,

    25552355749422598747591872308666588928648

                                                                        n
      Theorem Number , 123,  Let , c[3, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[3, 11](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2492, 13285920, 10857410096, 3339717211928, 546000062185758, 56690967587654725,

    4159112538163308800, 231252692031575705408, 10234644255282186107528,

    373744535472292946410720, 11573274580631978537712896,

    310445994329384691795061484, 7337934136474539387099350752,

    154972882180954756145444027067, 2958204542568733779514894098464,

    51532802988957709712258844767008, 826003651474008463059961830213476,

    12268069233351393701495023733381600, 169867221647990593021910769724754456,

    2204389801837590866534967696328051056,

    26936406369424318965774243499865691424,

    311212069950185987354742420194663795534,

    3412197283767763953156034056275680983040,

    35620487152257972711663288648497470636832

                                                                        n
      Theorem Number , 124,  Let , c[3, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 12](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 20009, 10748175, 2006225391, 198171344459, 12601587970611, 576464523905076,

    20355622768324741, 582194262577557816, 13963127400359510259,

    288214459121855475729, 5223877906802364604044, 84475427664818589417600,

    1234589333133187589619629, 16480605485037033199586496,

    202733801782465202611800174, 2315427473736472099106542195,

    24709589241343462355436188388, 247756770986799296998384143570,

    2345288744822172695409368120521, 21047759961506154662623450920912,

    179749473706414596252504380466135, 1465595895510265869804848689661941,

    11442606183962617377132385324171419

                                                                        n
      Theorem Number , 125,  Let , c[3, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 12](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

175, 294297, 95737686, 13410536011, 1091717054583, 60218980874406,

    2466576280244748, 79666296348135963, 2115969198821006331,

    47660802873304215238, 931922302719914114280, 16109942290512140682645,

    249829387609891555473702, 3517171173892965579187152,

    45396280266524201275673361, 541642732552618337871709506,

    6016161787151171280304630788, 62582852597281578632838748638,

    612893378990155349720341794150, 5676553469941840717582139887659,

    49922044140501705076345187602290, 418355298222173802546116161316368,

    3351288518898391640975973715203294, 25734533488805349349567943884900233

                                                                        n
      Theorem Number , 126,  Let , c[3, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 12](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

963, 1026159, 271640280, 33553560750, 2502739808745, 129289852345710,

    5029361409209598, 155751633781696692, 3993953813565024291,

    87304938046255610448, 1663337019647650053751, 28105867386901990065027,

    427135157724758802030246, 5905450435527034465302937,

    74986827311286215547669045, 881519220784282263091681599,

    9659335852789021112770809999, 99236336511244054857201631143,

    960736965349117668166866845712, 8803872987529399534351598059838,

    76660786707204702053885493485157, 636508755296676547442474240113245,

    5054809505197966687466599352410066, 38501051224630283503631402617069362

                                                                        n
      Theorem Number , 127,  Let , c[3, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[3, 12](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1544244, 90077883845, 544691212305375, 961804181282275605,

    756062281658022751705, 330692143238064493007481, 92151728082938728459909170,

    17896576890038653730156598103, 2580251577246291235264609039845,

    289322161498489339746579200051475, 26141129370244919863977089436592614,

    1956808448914033868637293403478132155,

    124082722266114269182567694941581804234,

    6787238396750480473472115452180560756420,

    325105458468839952499435226212490135423010,

    13809889077046235005404681513885898074441876,

    525838229568144197032417368006088362948809115,

    18113880133612265157993872874900630060657553662,

    569028309272222475232175742104462514864211650330,

    16415155836248480407540980184837946026903115819995,

    437527470860264738229179339205235249675323002798631,

    10833507420002711444817580286934338983001356617782985,

    250398496916617211593445430364998662147452552493422461,

    5425854193734668088525269277428443345256539735717167435

                                                                        n
      Theorem Number , 128,  Let , c[3, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 13](17 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1143045725, 2208105322559008, 205413828509284747211, 3706254086982911899936704,

    22824133923514893221668545933, 64632942312363949385423352789472,

    100817429967790925813870476139532435,

    97611924690108811745782501277853033472,

    63798504246441738428074639861284916998008,

    29940678059283081701831015702782830166703824,

    10574451360796462610574443074121682726431048585,

    2915826192096156283969125533719210742372762819520,

    646465927464727467965776224134360481995862856654720,

    118038139512272288407805420409224424641428894628937184,

    18105680893626904847686753638841325556588660441128584759,

    2372253894023861021748349056851746738221693648763497595712,

    269282377971171948703744391212465321756800997725503040428817,

    26805783218966483129221272241975051066679878361541112385561664,

    2364764120544410498512736579775884108117985380959977060454508817,

    186581823962522848606085719581363181085133542545738280840856485824,

    13273147270228393202190339479331741590314430020245578298356152371441,

    857429104274635459129027314387034591653425239745111280159645996511296,

    50617108686462779632072530297851314365810641191105483643503779410483355,

    2746220600802680936694008421639607817015349406887494873311476040792663104

                                                                        n
      Theorem Number , 129,  Let , c[3, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[3, 13](23 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1173238, 1632077847108832, 5604390339980079561961, 1419361114384032765963372096,

    74742759559584862996475459776231, 1332012854592498257325230016649622976,

    10590549135478039527497032152503516255719,

    44751895918972177026910615810969631299372384,

    113239537027043648464136347412709357954365708385,

    186939787244986996378120678652988130937643162123040,

    214658817577346619176715735092466745661134407798133375,

    180134631810729167476050483794397819268691232262693791968,

    114871850022867546688820673266098049731384400239671800559729,

    57450218363701761977887817854112024842931862561673300701182480,

    23124686133433340326056936245394812597071786503357172164663418513,

    7654829307580136884524149963145969452097540292533523811423797811200,

    2122153261370759631789245045603970930017150494422539898377512610597359,

    500431197820027779235308186218016512784494891240038706393632919713534528,

    101729804384752390999436751521910191084088287691257412550620535564505892561

    , 18035802306685411446182075011920266588277065557831159071618095760331362\

    779264, 28172430153802620226052844239871606534006577493012088648002710393\

    23335370062364, 391208867388570688285713927123574656307751207381462416026\

    486447778177418783120832, 48679005091118180355217718674391384449444540530\

    298140755610906739273738881611197502, 54664147412181014921183562744485754\
    22172033781742393647753270100513186383757096597408

                                                                        n
      Theorem Number , 130,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1479, 2354398, 872388762, 145099445649, 14186655807771, 941450624488272,

    46300762039355105, 1789303956082022566, 56633764960383511810,

    1513775631424531635353, 34979520696476288250118, 711745120526056995795418,

    12942332655002968224103992, 212877038215054900712795070,

    3199153846341339062732503965, 44299927492870505330724526219,

    569322681784618058606236775988, 6832659615349293680761521877916,

    76989088178481557304546709042429, 818306471977360882598242354894325,

    8238458223682785092451773905266975, 78851558113133802188823722201161338,

    719825260731283724509944748660417808, 6285889169317845631818889961209823769

                                                                        n
      Theorem Number , 131,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 16110422901, 28790121200618072, 2977624805491001868875,

    62610893592663206091998550, 456038135034680977162096485852,

    1533156653509522342891251809716767, 2837299108768057177794051084002499746,

    3250804378031928571864912494209588655512,

    2505777985587323394450016723931506638618375,

    1381751049110264866206520574335747287361955580,

    571254174460552885121713411886015408225809860540,

    183710341693743485409848208005821044433618435658921,

    47333812592281629155102945871355819966433653061463042,

    10009765329680300800869167577684453854137318200291213745,

    1772500695946449967475641723297694289537736905213776115168,

    267281836959211103243189402507897716260995659558242267499501,

    34816868442624079547312682648670968437956363607679119851123458,

    3966314282213477999172205294074151646702206790685712393173791840,

    399382745362437038619796060046432350219792265390080085685485646855,

    35878825346757223471810837897310314781018541357335194070069011971744,

    2899274470745470970871895288930348011726673697127969620745288167792676,

    212270693990681808243409762665116372248698735689852580500879574298353485,

    14172372684322050337560402616998708481841512854350216410177370701361238145

                                                                        n
      Theorem Number , 132,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

435, 281649935073, 257761295343359938, 18977166558053979643381,

    321551343526797254712888874, 2009605769220380494341624081740,

    6015969722097918028604001222963097, 10155285757053130121478907731051687251,

    10791967003886902347009502689112620466210,

    7810077174875347327921928438974543302975475,

    4080678795694042823935986157225122809640372560,

    1609981809275155515360585693893384425227321299938,

    496913234794112624143140597004900920681793942698575,

    123445518775275151634102764319831615710960173841320712,

    25265955240729092712444735861963982927438099941730369926,

    4343976488949000908002942638077423856622042801541739371720,

    637718863454129139905927951185263083004721054178720670413306,

    81060461068312019652139370074857442531018110730435593145333001,

    9028807555501119840674265089790248826339384110286558889345115009,

    890449283611996560778719028929839679599251360770141008513153597609,

    78468443984738948257572059360534219435934836653919033092276382276795,

    6228252242317649690351063029312234947219769049464152301673807245298782,

    448440616305602468360407433347389004929835136103300677154009525718392477,

    29475296033688466458597060092156522349908108870001539995339655935087984939

                                                                        n
      Theorem Number , 133,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

46159, 4172545825815, 2142580009968792465, 115673993890712341427654,

    1601643098698004474189725990, 8656953450587545478840444624060,

    23191634739157000643771683000341570, 35830552626312610829764389421630008707,

    35402635986192630018601289200456356541123,

    24097758187564287283453752602729025149537677,

    11946653167292121437337160131465701200638688187,

    4502972963617364675984831217446641160788752638560,

    1335049847730532366286960949871236705504632323505826,

    320009821321786817963923903465637351133577998941570581,

    63429771059657244874533267057098604028081218099585478742,

    10593866480153662641517703702167391825005017985532993480726,

    1514756724794098784830740807588831425916417252719379322048500,

    187950679314293915716396489465185730176313771607764808183627076,

    20475262066786889909319430788825427246462671198922819895145149479,

    1978370560867891881788184978802711567781519644847822716128401326259,

    171056096595754161186734544936164562195379802846467102720609201179314,

    13339070996009668679080965396570359554010277019977368195508569383722765,

    944686679833245309522195875781452861062527212949122642551528815302390988,

    61138990604603119845364224366278132291977268088933977712813327107321145549

                                                                        n
      Theorem Number , 134,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

692470, 23272892541215, 8467664360887556735, 377143143382232013247933,

    4596535996311682788000383818, 22643849464259281560160796777185,

    56479335814325311583771027225143050, 82402439707575480615298172747008992121,

    77659404404188842589809806410305098381974,

    50793657051184531618652732423601892328701522,

    24333029973836337428254583902748221755706446050,

    8901858340019075676290210318313251726971461127422,

    2570656012310134840269135273370248291934964843002437,

    601903111473102161744950441631054206739578725173011135,

    116818246265646567480284926576956214876563822821040412249,

    19142460306820748386097332340010159595467617256097494439875,

    2690000779710087416063178127692513825488027660411554872255719,

    328515103793051942616843238583912053964197622370951226527574142,

    35269018825186982417085680687158771574915297630079752760074448129,

    3362053566108523344228455166691445140113152281287520522361845765341,

    287072225969626574339017384274342357215257857535882303125149724322375,

    22126245361299203182413421565370765647892982227246034084696592616954245,

    1550001792796796073415924674145454913499775531176775338543623642853696322,

    99293915642414033901279408706758213229921050957762081156627278173629364589

                                                                        n
      Theorem Number , 135,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2461919, 53819248428338, 16656989694230444650, 676492154508297921701335,

    7750623631922394566175457893, 36493467477147344393601823471867,

    87894722069335116382642072172913295,

    124680787323803664730616697587795128955,

    114802257181200888049687894672871075410755,

    73624806999437287246289119647485325525476398,

    34678740245741382792608673181834642566645378183,

    12500806861118424989499507508684333975234968387555,

    3563233441294830434261994700983597873623434260472980,

    824679142478734315100821475323097840812211565175498130,

    158393647247798624286806985557346959931761203016167171805,

    25711227316689358607174017531418480749801503793904872181759,

    3582123867540144940157643488515515513798945080407311313020055,

    434030642396670372694125262137693369751239516655548598824945905,

    46260166288269561891542838284503740333405205817561577621377420107,

    4380308774266302005690966082105221388889546926649908778993963860040,

    371694823864835778344287292348128761547106749478885337414765107031544,

    28482839858329516060878779534769016152004567075080627109920004218729080,

    1984507320146016282484987818579289918905044169577130116206388356621942850,

    126483976153515730057357738585889614034923087706831266985585158830375594452

                                                                        n
      Theorem Number , 136,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8345685, 122839070381074, 32546399951785956007, 1208255038099225243002970,

    13029390439079560391389566535, 58678869320173301060524309303610,

    136535530728462939135128908472277656,

    188370457620204837852900837419401924540,

    169498318637611512559235345702712688425736,

    106604741850345450804155146684125373445300935,

    49377547835233524517484608120622734849598844043,

    17540555451388776966957861230807161193886727823973,

    4935511069503867276740548919538887718102245380464935,

    1129181203795031357346133126402585818606615523586101270,

    214640837433255798509867772869036868904345249699585686670,

    34515843655329009194418918367994958738388423484735138481459,

    4767812814414551050870885933942657580010366869205627668981782,

    573182490989469873485800722109631440690598502480981273808829169,

    60651749863206594357924459506009250422800654457976583935881268195,

    5704796918620293411501261544942263091876552539128757500761345417115,

    481092648993039369626541779470542002545705050010158476282035558901390,

    36653541385580340971344627026684699166432933840333888002172052489857412,

    2540033379164451005875774571611302466845426537642343644898214182434863774,

    161073013609213276044890525016121274572085557818977013337418070612598664123

                                                                        n
      Theorem Number , 137,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 14), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

256584930, 1358358538825982, 233632595867004183244, 6715706538701544256418184,

    60814350253624253441732990377, 240678374110988494881835986733625,

    506349755752903319775205182708866139,

    643940144617130802334382946723871738118,

    541519498408685033739573854666280824126521,

    321587897990605773666262533916887100245849613,

    141760178203643670319112551134151674520031778800,

    48225001769043886028765341893299028151362682005178,

    13059958705893048758127336028735233590460209211156696,

    2887607086271908277085219511853528971408581157061510546,

    532274716602790394533587509187545946539413955481132740224,

    83241858839580087222486858195316946661143446731564467367970,

    11210064174698905817228881402364724783168025990884753344478367,

    1316632597889509645716242521691533350144978166075758254122022265,

    136362163802749339836512679345765024911947889102073096424434922546,

    12573781644328255001979230041414970983458294153288098477923556761746,

    1040981224602949076915263077521822696423013259759355367723556901212285,

    77958248491892727144791231453002200325595317883661299733634769929701083,

    5316207320101792572917565171162297119967985015338459905881050201792070518,

    332074526777429619852016746730711152100825929323467767416828552972526884619

                                                                        n
      Theorem Number , 138,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

752093280, 2959193909979782, 445228126889568128045, 11801766667644001472317993,

    101052451071997136779238289230, 383578241095798643924214178206325,

    781033028632676104426418999998770504,

    967260556079517889489130478970881762300,

    795637521986579942786165690806441840765850,

    463705813581394236462264160036449055328682173,

    201116728183685145492280193946762968277295482841,

    67451957201083736106747763982779670520751209511835,

    18038505836669152956058300912455524521193484844940590,

    3943799193813928993381250892482029165194340701934071102,

    719639552571407545254635875311127962238370398148210201279,

    111514760332140703903766967061105269220449556917028789068745,

    14892203219393693674478251198179506843955866536737193375419544,

    1735706375152580332842626772582282778888685272789694515226101875,

    178495570288874919959290824759348782796620519648090710529293739249,

    16351212201425467917652081409687638439532713411149122085951907486874,

    1345485097821569960383515397289222335496299584952343582588031039329867,

    100190997463288151019032364557238949131027838247459292543896106937779688,

    6796080484014094294244124890591729326631135511765686845703523374586324350,

    422401034379167610924504564858989620521985935895105093320925692622152826107

                                                                        n
      Theorem Number , 139,  Let , c[3, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[3, 14](17 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2142686216, 6379843560408613, 843546911358865872078, 20660249217243594612445747,

    167447847583711193707716698820, 610017864082834409091203934585620,

    1202667247145156977144482652278112201,

    1450873005655847685562981842362507492881,

    1167617240252880459339431421637324386610143,

    667948035586622740762823737524926796515716085,

    285073416148576707520129828584974870591890099057,

    94270819202429904613415608495818762416275360411393,

    24897578490758200769731892179568097274329643604765037,

    5382948508137808977635533958559723140057093558252071528,

    972408891035520489687904407817398475656619136610077198680,

    149313715750224055544896742233445549714236117441280078644202,

    19774499008174696008034003537926163874407007990722204199792263,

    2287178019840235778494850497387575742330731455743588150115091234,

    233554018088385237833465022055479173622057609143206877573801471693,

    21255576676135489871465909456084567966297619206576374213452137726395,

    1738460743815920929322641800148823731771085507115683202035109385814272,

    128722719488469495287747355554084639783548647059868243146976343675434349,

    8685280139266413093686404674406987000880745482323899996606141968518693584,

    537144379359493726990854247545960523667810073381688028598657699287008343522

                                                                        n
      Theorem Number , 140,  Let , c[3, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[3, 15](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1287, 31778904, 75250991084, 55396277624574, 19388539066249854,

    3996228622709814640, 550476410667582875796, 55030392651700603464048,

    4228144432056187042530441, 260354822585921719624085112,

    13262384979327173716201845402, 572864529394009432012547781232,

    21400338265965125756827013496816, 702576493583764242278373351459552,

    20541370056038232025554658280123824, 540822555585530550330577296347675088,

    12943870983471414598507771732943867826,

    283901074066784853897317902829229430068,

    5746444516772972879022591065206811176320,

    107996149940813227537771997908423305551760,

    1894610529943274876723289983994815580535579,

    31173865904868203849923884705069443324536656,

    483113819640719651382928131484752292817017116,

    7078347268835205726078061272910248661598051184

                                                                        n
      Theorem Number , 141,  Let , c[3, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 15](19 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3344130, 625108480236240, 778810658062079615472, 95920197794813265124662936,

    2841319813040065038347807790294, 31120503451237135740059954304030288,

    161504787602742389031436066831472147034,

    465461775016378834485094612335381139427808,

    830805775910455395862466371055783481057671336,

    993695682607836514011590963133637443406397744826,

    844977626062573378976722075084311098460957356178075,

    534769434332148600354732137706945578478306012705624152,

    261212372911245146560467224725691265457201962401732715894,

    101412847716298182158958632698480000370936832132214041225408,

    32060514738604245405780178838017579682995520803136639321837193,

    8421656646990421789773655094577905280485284485155584627010148960,

    1869769267049035100715445208976593351281326454794232016190561798528,

    356021674100947759294200445572783273645608526502743405898305932171152,

    58874398713748474934058832703463763183480300609530719048731789711373487,

    8548510258933786237215319870895511488588866773477790795795394032663683024,
    110035502142065164811339489244352444876973786081957063675621759538063517\

    7300, 1266274111243793159409700639302101943602859955311723349972840124336\

    22296767584, 131261148594243403304816533410300959651807416337951974136959\

    52998216196279703003, 123386800864619091936579725636437643263059147437866\
    3669068232018456544870289208632

                                                                        n
      Theorem Number , 142,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

87, 2775953283951, 19353644440768955476, 6766819631875180235673210,

    426725416349738120672414692427, 8499099706295069875725116282828762,

    72641856261039626087582702385112730087,

    322153985110526618122664905546004495571286,

    842126500142433644980911692008764468963783515,

    1420631495609667545640476424698943713310876209493,

    1654066598617942492568473363095068829652048965570825,

    1399418165289736774419375075657586582615191263291727750,

    895880079489605012807012821083305017780462877474272121101,

    448325194579768737971710691129497665303442008036985518075352,

    180111578312430222573690995325429879483719796467064090631079736,

    59388542568578786937696695885055190799179548554707181362336821547,

    16374290115520591563655170375083679071249655782257570822422748466130,

    3835345078814375253717156260528062860126556626740705681693008996270145,

    773653244176773309076481735075342631969573864630192069024987183998576305,

    135994261385769099350239776519517129960148175205998521609307985307771274459

    , 21048112681980188503170525945483994482987818413166720312604953311335315\

    668525, 28944935978596246212397546856044927612203765946191253962732728739\

    34159130349183, 356530098482299647453490131428399219151838394847300781289\

    981810047516308443395721, 39618619898852681966201584001273017122782912680\
    197274431833724053216919455252178873

                                                                        n
      Theorem Number , 143,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3250, 18144956245044, 82649125849206707548, 23174696747350318976304994,

    1268707066403873893770764896641, 22846343293521027326740649029484573,

    180864988421729624381236790468587141593,

    754749496810074079940972394177118609397048,

    1876957627915847942055922370451492346756352113,

    3036503975862183845590121825936077912923391288124,

    3411076708662049507987195702247713862464547560319381,

    2797565196297738367789157552152763424792518646222224023,

    1742644772434954688126595396567474750766587376465067149955,

    851146357576231852493112314432282472132765865091272759431925,

    334579662726184317373144193065826655176582443016545059233599661,

    108174242165409039854222378614531119559663712242099385818013253554,

    29297022761629756477652761719972180798512234648529565323240877265509,

    6751031479217202210192216200579847495317018209660016857382604201767270,

    1341501772064743730212890897551211056059161685377168591946068157353121511,

    232565472920007746514651617848305315563719667531606485528476926699312918001

    , 35535043972139355002491097425433992889399491950732861617727036476896404\

    026569, 48286240279862549453109398007772709132200828936344722795725427209\

    69298307641079, 588164688878604204916038218640156314085398435505512230141\

    184530221691752338951428, 64678829137042335218031952253438838937000531114\
    081885104131612160542843938558340121

                                                                        n
      Theorem Number , 144,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

76268, 111215539796827, 342780917470512299533, 77980516310620443601149179,

    3726536520609821670646312763243, 60861924186324547800231891349799043,

    447159942651482010040906632691576131639,

    1758185592195254503546091574108076533536025,

    4163620877160444061727678384992419110120471382,

    6464232589909014054952792355550475298352171183833,

    7010035398457432159884315096601324990722441935233624,

    5575608507389154543547406294133067281001587208319111486,

    3380645643334561187472452457031120985178365295243800652275,

    1612026537155743701889410718551326319574740089358862053442205,

    620180901827039712673895438671345887640201939477278955418154341,

    196649916910319898009349838600007669544659229547495536298950030718,

    52324854989813655404232234535538036849129112154479586405469822269634,

    11863798734818332254430514143917751203498490660221784002612333767762165,

    2322629971776871552364645905221869228313784229961759316827806609829989369,

    397157440155113307657571810346388652335077405428899933177665331339339227957

    , 59915124832688707637031893105513416164200347257548737893658872274049181\

    395685, 80454124744550545882556372711784732646832222689063154333314862484\

    69722280322397, 969192141143882765890269890095116567338749356315122910646\

    716282984749176206652604, 10547843152844609227672084334968743982722263149\
    4419450366514560629581765552488209320

                                                                        n
      Theorem Number , 145,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

326858, 269338403800080, 690856011644591012764, 142136315902344544414018177,

    6358540907534556747435195286713, 99009327939564867633770254692202927,

    701281502880266551722757534714044443319,

    2677826404474090518251278749315900520232879,

    6190396641623691643132690379894626363496962758,

    9417624519868033287531144911358847643905576711588,

    10036338590104691631411441520517975845721741006375765,

    7862462100750908819770462686971364111647386654377086098,

    4703936523266925079489730656913800531506771837398219995229,

    2216504851903343572842631166418068616174965446163103057575394,

    843681992120916625605326142823216082262711837506273631221910826,

    264948904999165503680939523870429411128695701591906150958965257753,

    69881367827998545302166530868373274874903824574061505299275962605067,

    15717574559940918018163086935425149052943049685826093327584805069451760,

    3054429288189659845523960520612459821788103127128989052990041826909576893,

    518734625476903334964682950496833402540734358050419679599017772478186264887

    , 77761757689099765927103168924686046478498944302425534760310214647090793\

    564835, 10380415870078118968379690777773568930731487357087892824016524584\

    466879518541560, 12436036189090239995150061407771101551621307065112446055\

    64326563059663011752109129, 134645667223283773681238286441573802961172851\
    715486503771447224296195598706354230923

                                                                        n
      Theorem Number , 146,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1312376, 643384212472607, 1383139772635268989320, 258007282059737701645029289,

    10818278323988879240899625736633, 160719290881788564523130974499185268,

    1097952920977779369753434782186136564138,

    4072860237907536875841800473893814179352543,

    9193152304239496382359941321314878317257398854,

    13706890256243160170543177054603869902159545379858,

    14356916065492250945262734562739519056635125748952066,

    11079017738425343217494862453883576313667452700185487715,

    6540885563875528486830960562509323404032671628603566728475,

    3045852184020098033406640617451678178935310021261068606324931,

    1147118067654486273298554143960773590819254574342308962088764597,

    356796693687297326608238394969869430829689047333734892256173764611,

    93287477216844263835736872935272921946956258469558051251207571735523,

    20814769919792494705004034904372910636197344499872674002963744285845952,

    4015301253339921445763259133670370175636774462438467470123538581074647293,

    677294841576242216372757962237807975988685652041118293127265742046468948472

    , 10089188415106123554189048788721016937121821834375318833250624296369424\

    2022814, 1338909221708692918191732571042265107468121832535239630231614071\

    4195103275893340, 1595263314734320960463045810676278087583472355598676591\

    045802341896952756977496151, 17183313433374083960930694814803185678450809\
    2025434600317411285248273019657213183126

                                                                        n
      Theorem Number , 147,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4980027, 1516928978313096, 2751290968495299831285, 466449445438499580126725895,

    18353888290849454847612289701625, 260335930429466953664948353585817365,

    1716111157387447443260633060495406581517,

    6186172250356525610688571500940211957061686,

    13636834348044513593717581144721508612113773675,

    19930264732104036134147837049049699489276938749210,

    20520146074804297243986392248121463833864809729632920,

    15599924168628723897384731041477975722869515473334570043,

    9089221574998902851919818419801987954895133359272467414795,

    4183059515865323005907824731815454833264522517496248940930311,

    1558861838712412825421539003090740602675292860908609282859788633,

    480253874081368759586523854760535788046696507601666136602609613185,

    124478622709275018186241706056731856690804121618729793874481513201258,

    27553882036224308365019814608825399799609095520746563105972439000077750,

    5276487008562445688715566162160896190627856269865315088986539448627848268,

    884017539576723363327104228468138059939038439909318455970731465025930353975

    , 13086017049711930172399375066750994337280529638518416297513720576323644\

    5156733, 1726465554735090925405953023734118624302142218236558571363237243\

    7885111611014393, 2045792122416460962808063570310301525783070319218111791\

    395537059759767829115274112, 21923385674074297147631108880269327955773376\
    8344788551854960708800455050398114436181

                                                                        n
      Theorem Number , 148,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

206426996, 18488480232203386, 20868095184682279332855,

    2692493492153292594315001025, 88155942715504777490912059183090,

    1092699829624917389044392694246617877,

    6488367449021394218084989441753281524111,

    21502094876825539217004342265731511229369221,

    44210275489644667340048162867908553827097070868,

    60915552685384502915980968694469869413592652243280,

    59616306731885607623737960303412573048362046299643685,

    43358916063139191961537212893637027329339300996407049724,

    24294308341838156371571476636307267968038330558403387738330,

    10797779865320662995740790289030101422102795431253476403042373,

    3899771152895564945870230622969804848683212511720359132747827442,

    1167829453389531177579350909382157091394802284193223493863307051069,

    294967835684328093660264356676011006869708990163685632656011437107824,

    63763845028538309626266209023147778249356790498012010332714719812713820,

    11947135883673523617119112808803205421907792200625827841595598472484709230,
    196164115189234384756984787174402327601351403806758232540483858138671283\

    3297, 2849921567869039566054150263527554078621388420103994038299590223909\

    15589634447, 369492226968989701034514667602698553265889575865110292614856\

    20033351690828999700, 430750041518951193105663001915595002118620933737340\

    4775495112756923794354063237629, 4546011809479944473243922337106178132583\
    13549589227832797776939215347935354003633701

                                                                        n
      Theorem Number , 149,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 14), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

661693580, 41601683400169390, 40521571762384872800867,

    4793602631906476719103637845, 147942882489851876397810881928257,

    1755438613481193266133956396648998121,

    10075359685142614544821238969586391442290,

    32485026484148618118631134371421069993758834,

    65287205020251354841336748069433571551606784734,

    88235163172820036615640539787384192417386094769165,

    84926670590781152281338520961634813264933751173624088,

    60874400675330266642545571252168607293964040191371100785,

    33672234660055115944455194311290455386247235728915448404626,

    14794974444249044590590397651128608941637336962493847792923120,

    5288484391020002496316319234823676890541068109864087550867612152,

    1568935507949410202165946043026987398210675467712174146566436722451,

    392909667340364739873468671599225451702890813925155692373775779734390,

    84274003038094058467307791787529672659538258190034836792278139196480900,

    15676599856632269991860278959120079809544250447720806118753181006040859870,
    255688285290910544148929382035962328262735743361178193725110019313688641\

    5809, 3691761607258661133385401827918120290321426741211263824147722973334\

    19505435700, 475880928154747873763709083159750783676555907273542309465702\

    75160188530332505425, 551789521248216276528726227193913222703958902789224\

    1295462841954446859574264677498, 5794016068364461699679612684759433184032\
    50830449172649639983201597668395393942121775

                                                                        n
      Theorem Number , 150,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6185256875, 204202118456838180, 150254923727860720654433,

    15029857758233818102369325484, 413418493726165181521924644245139,

    4503681811117973330554576623121688817,

    24179657757071807635223962012821483643611,

    73857709964474391830514705291163288230401130,

    141910701197413310588315591867271575493406109677,

    184608014025432613850204004017139493701976300741846,

    171926770649346553900508360130358633543458167780435401,

    119733558562081227418733250399695844827141620709854593387,

    64562284783728761733604269580614193248845230133287341982785,

    27728822732208116061512489236124093025287129731171090491887207,

    9710572561061784716723442392669388527991256270657251116128332913,

    2827786488236189932525012530323735245677121818104200002994181726386,

    696259420163308249427591412398621785396982624849568148333184147758305,

    147034470961692063264392664579921461704175431870198756570179674254001732,

    26962151532574971818117453757532108779184574137691702554942652260717105634,
    433964648084783533169759382514414910547522829428141079673792533540328939\

    0276, 6189086706672235855306874516215612354670631413918135167387707674363\

    95034605136, 788683562983739782861404884558357737327037687016419315709591\

    86105430090695375529, 904716234271344687908736442920755898060504248544194\

    7968335115025462805182221389982, 9404644524423459509240749669506402475099\
    29948713895941424429356529945798348451695292

                                                                        n
      Theorem Number , 151,  Let , c[3, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[3, 16](19 n + 17), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

18129715939, 445808751222034287, 287006604871282117681275,

    26472852184856376991951898003, 688458701662788608060158076825605,

    7192618520852342099659255692877238375,

    37370565715489516956010624333811317735050,

    111151882536950528232642925110412671855546218,

    208884133183732379309985787950617869298550706802,

    266656480683946385212420466813328422742473548612794,

    244325510406499016234446486224399732867813538746168703,

    167743144608987634773453198027093752474585436182714024515,

    89314446314836873426204310657968457544413648423373887630531,

    37929019426904575603572204375549848177183017437226590997418836,

    13148270807062380869448788441379648693733436480272218923232952312,

    3793712613743220920755796055509917595746536434326578016311279253092,

    926260378513862148303725407724250129893397913987387461783039865373559,

    194100673326976351113400293439130215709988167718644526220579388039794297,

    35340316067846107567009724131560214005956886159027603425513050220928651634,
    565077137319855210914923424007845625124043093409304130368954260143110551\

    0652, 8009764470321502457218151834290543515127753940277565127398431122245\

    72074114406, 101488113373985813331180276281870273807013734381953672570014\

    513268233727350230218, 11579878364719920515070061438808942204933680393456\

    103629248791367516836761547703839, 11977237438873389654703342014389638192\
    67289965086095189267739565066003069924710970775

                                                                        n
      Theorem Number , 152,  Let , c[3, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 17](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 80564, 103025832, 40670309461, 7870799093796, 927901429852859,

    75417898023785128, 4572603855311347752, 218284888050701798600,

    8530808248115270355391, 281048993441648647470420, 7984527652272202133034250,

    199163804430008402502778548, 4425871811359201986220837389,

    88681335093349368931893447388, 1618346984824444331422444394734,

    27127360461993347984323523969672, 420723410817823243582537021200425,

    6075261488277882232827554923926672, 82127018248850544568836815668046784,

    1044340818157201911534992379224741232,

    12545056182004215918323501892568178847,

    142893058191638796287111487091459072148,

    1548522775844367946024117713037819952192

                                                                        n
      Theorem Number , 153,  Let , c[3, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 17](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

380, 1707906, 1252120524, 358530803833, 55826535973276, 5605869843874302,

    402094211781029276, 22034512740355103028, 967048813717621731628,

    35188662490037765095255, 1089962164275709846473928,

    29338040534884814326003771, 697641851482252142234963480,

    14854760603597894567461394834, 286404107057711712761986913128,

    5047158328182703028395721393122, 81946695033396310845594181247480,

    1234259304764014915222210623310769, 17347963745902119732329307769871188,

    228722207951307970572524549164572975, 2841618432827236488722168502029651172,

    33402093098787084702038951758466869249,

    372814733438781992886209995360606326448,

    3963887420069811011757812465799026600666

                                                                        n
      Theorem Number , 154,  Let , c[3, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[3, 17](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2599, 7077956, 4125553125, 1026041480496, 144784931759901, 13501149744217008,

    913443201964725402, 47724560010254134852, 2012524153736787248285,

    70775517633846882072800, 2128348723209782061463694,

    55817836402389813969375572, 1297026037740194567728089973,

    27051889875750392440959331700, 511914843718337352718888824681,

    8869291819641599200342416294484, 141784100790998685613759104496747,

    2105237679034260978138052188673716, 29202235958023067208329013415526670,

    380333776642600946505407750046137604, 4671741329123168819481686853947096979,

    54333756275172808317822626705682769172,

    600432087924941467883992387661431689926,

    6324539109196495827549470937981010818788

                                                                        n
      Theorem Number , 155,  Let , c[3, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 3
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[3, 17](23 n + 14), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

896859375, 2192132277167435752, 22975174447696613829240383,

    17814498920303545835998735053800, 2727186126522655634418152585340908445,

    133573666846071196055669631243487744118200,

    2771741400800789075844652386497770545509253600,

    29196847959429124054637577738013530250658900992816,

    176855840270138133285102489352053638158871933817532045,

    674288524513406984215851754623069632953007834199349156820,

    1732010550391542362964209278706989721987858693937332635743455,

    3159615754124943683618320710935484860017034178801286493057929376,

    4268597218009834104713352868371052364099348483380357293731752028716,

    4418128445223124684337344524493557977842374188485642413738582833403120,

    3602732275106959209212750696741192530741139388071866465041904730952646615,
    236923547212518719561420777338464292439876725538049839193992474098646998\

    6400, 1281579979245863458049184264403542942201136459379733086612653528288\

    982092511605, 57993402315119559876643388412643276678600907851372142555235\

    3591005427950920134768, 2227594513596073954388925204802657093167674743280\

    91036540205033888282473814961836370, 735573590622524398444789272283881819\

    04816626337710053776859407881550622300790792884000, 211143222207963645307\

    15571369981133086339595826054152721068907260746689154125300089703220, 532\
    047843409605015294896696662965113299069818021348356389996708491991322875\

    1607484235219920, 1187218030270856215558617444947250270867881368755821843\

    231494203654049462491609861696259248784, 23642558545617870802476076152120\
    2028824540786940825280606061438445363236936294750286849287283880

                                                                        n
       Theorem Number , 156,  Let , c[4, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/5 c[4, 1](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9, 448, 8736, 107072, 978976, 7274541, 46200640, 259106016, 1312555360,

    6105390816, 26403281712, 107186952704, 411623695552, 1504663010464,

    5262482784576, 17685687222389, 57321035044448, 179729688762272,

    546648048085472, 1616573660106656, 4657743360212560, 13098942810164128,

    36014537731027520, 96945093344416128

                                                                        n
       Theorem Number , 157,  Let , c[4, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[4, 1](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

79, 6240, 190848, 3527648, 47015776, 495372704, 4360993440, 33267155339,

    225576138336, 1385262438176, 7814352258560, 40943596460320, 201026385196288,

    931582839518592, 4098997213935632, 17210357417631232, 69246495246011520,

    267964970814575776, 1000447483686337792, 3613563496154945792,

    12657458595632896640, 43087361564491779555, 142812644683111929760,

    461668813649783489504

                                                                        n
       Theorem Number , 158,  Let , c[4, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[4, 3](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

91, 432448, 193162304, 30270442080, 2514830150560, 134698091112416,

    5199739598324192, 155276420193745216, 3764594553565195616,

    76720920091932724320, 1348917693200603439584, 20876005343458687222397,

    288930360978665311083712, 3622336527629467941156480,

    41572497292934925230865760, 440611275529859674082471232,

    4344634906354901245674938304, 40109132869546929571824817824,

    348569682855537645447189820448, 2865142970492078430248815977120,

    22367194482121633732654095979744, 166444796255461297237885571856096,

    1184480853699968465448840178404101, 8084227243972469907675849663261856

                                                                        n
       Theorem Number , 159,  Let , c[4, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[4, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8, 2624, 186528, 6588752, 151384032, 2575936832, 34980502024, 397673038832,

    3911160234240, 34078419498368, 267791603622384, 1924237675142272,

    12782797282819328, 79205071274611232, 461119439445978600,

    2537801946468964096, 13271716891795032144, 66242579733312205248,

    316767803167543452096, 1456051386978023561168, 6452158181375797773120,

    27633670846196357751136, 114647687026633274975040, 461707661786650904933488

                                                                        n
       Theorem Number , 160,  Let , c[4, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[4, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

32, 6664, 397616, 12696552, 272529088, 4411253568, 57625149312, 634950375200,

    6085135994720, 51870228076704, 399972444293232, 2826994714336368,

    18508035870998848, 113198035641926288, 651356232534368528,

    3546990754017842880, 18371031404703123104, 90886138500364077024,

    431081041388083803136, 1966606084337537810120, 8653703714679857771840,

    36821275729059259601472, 151835425004106820277328, 607977525014540852908896

                                                                        n
       Theorem Number , 161,  Let , c[4, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[4, 5](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

769, 9710867, 10501642178, 3602033513003, 607824853980470, 62467083048182539,

    4423294523416997021, 233566313667226037815, 9710800416417773178088,

    330636314384223188914847, 9495549329340864130729560,

    235337425061456849557935373, 5125517130332234744579799877,

    99548032002057135267895451600, 1745100788612854898792092990606,

    27892004102715987872072617755216, 409931432596473240773373030178537,

    5580550388597449869715218007571247, 70811600930223803377578722486996165,

    842103380588122970820920852561047469, 9430584713966843288439984591160612693,

    99875219678095847460271546571266576757,

    1004038323956265553461661604074778749832,

    9613215810402397844540602009097963384913

                                                                        n
       Theorem Number , 162,  Let , c[4, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[4, 6](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

58, 24325, 2535946, 131133982, 4335959718, 104255104739, 1967108552114,

    30608532779182, 406583471702534, 4728334248916771, 49069861974445758,

    461248497692423553, 3973975543624591154, 31686803684438389906,

    235700080469099176084, 1646540756997807749981, 10863724310401062262078,

    68028066369706568796523, 405999039265680501180414,

    2317821031835844278919813, 12698417339762739540225928,

    66952806733658677415087301, 340592704289467753420743502,

    1675447054851266054406017452

                                                                        n
       Theorem Number , 163,  Let , c[4, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[4, 7](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6367, 165121107, 359006749466, 232205778521928, 70080010290785758,

    12347789850878413194, 1448364283729708781905, 123124077805418785133177,

    8044564912706730255179859, 421595550992984780868119553,

    18302237674208935689240824550, 674814424128509604035738999659,

    21556134826623564490755523286669, 606270683975425962595691397994580,

    15214070667620817379840760358131682, 344457681591971117375485551718896146,

    7102723174565945977472815437328370116,

    134464836381722144719987491838787330571,

    2353457198398868551260204539633735241177,

    38312984916881232620153643588608325292612,

    583220896645035939672085682023105752297854,

    8340708534399047280214005812543746777083138,

    112528738499951139306499734136183900188505624,

    1437570357147774922020832048348721293136804565

                                                                        n
       Theorem Number , 164,  Let , c[4, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[4, 8](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4666, 8533696, 3288936875, 545930096256, 52121936860894, 3334271281584756,

    156778648064334893, 5761072496258864256, 172737825704857694878,

    4362458436603905559168, 95070096732305647312274, 1822030932528210872514048,

    31178720941674089561548002, 482313680513683938969006656,

    6814386835790384232198909600, 88693912336587399298903139968,

    1071302622746058540152608525758, 12084063865119117177320658968448,

    127985858625385046428061874939431, 1278864387350667494919055267525416,

    12106446100094467183250811596467428, 108979731974036350590515428270642688,

    935927612011416492138707638835923182, 7691062209035985069391783853565918080

                                                                        n
       Theorem Number , 165,  Let , c[4, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[4, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20, 34294, 8868546, 948691760, 58657362818, 2467304473347, 77603442925443,

    1939608413417244, 40173940200551721, 710867878741567093,

    10995246204493740251, 151328874339884867227, 1879684948304113105131,

    21314699241272781937758, 222748491246848168993030,

    2162369731542630073181460, 19630706454053439192171018,

    167619283147402258698360625, 1352838883585244795074198589,

    10365173086776111174848129347, 75676326763920719507657603514,

    528261785876719578991767539247, 3536183790246819413629028375183,

    22759891914926786790624047503017

                                                                        n
       Theorem Number , 166,  Let , c[4, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 4
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[4, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

117, 115049, 23875626, 2246527231, 127315444081, 5021724320003, 150230729925115,

    3605846988784335, 72214875735762783, 1241857863646465355,

    18740793884032894076, 252432141781751171411, 3076286516993486634150,

    34294490977113972428360, 352938848125987097001630,

    3378894987228166012941343, 30287934831337622686881258,

    255624241516068816895562020, 2041097884435423211464154861,

    15483886407472861246104769705, 112009266462161012124394488757,

    775182313921444875205989040588, 5147438956773626052195819601060,

    32880940914046539204170182792626

                                                                        n
      Theorem Number , 167,  Let , c[4, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[4, 11](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

172, 273325, 83963527, 11132965600, 860088913219, 45128335270378,

    1761947173582961, 54344033619350198, 1380619134945043845,

    29788414800986682456, 558674496002837356995, 9274377031168876840664,

    138267161580760648083288, 1873215888727573811147555,

    23288070180179170103395872, 267864698797704086136074608,

    2870487785699106630153067892, 28830113821927293870030934635,

    272791797308111445699151931180, 2442691445899435861386005777459,

    20781614019860578855839162933301, 168572127697619076891070489222872,

    1307804538711105677869288029045938, 9731110640977566294272396648689750

                                                                        n
      Theorem Number , 168,  Let , c[4, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[4, 13](17 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7607057, 95409999021440, 21966303778267022509, 718648897507165753924602,

    6900518521835396876448455975, 27894585540498866582780909257821,

    58636284607668222640855464754952680, 73474715066662893868648204280679011199,

    60320088020738917309254351344420010116044,

    34748867793203693451291303372169859150102472,

    14792601301937150379598818236222165018726620950,

    4844196736303895453479126321928711139811524282462,

    1259930909685274570400069133069186533899769582750328,

    267098282987275480766185879735011381916963418305973754,

    47148904685439998730069659577146179352225423002252701138,

    7055133414328505099722394703722141435072167792132933417499,

    908526221447231004972280409572161805704970048640304604228958,

    101996337143221060562679240543011826451366562731485334805266270,

    10094756057916114859627081097628251214224866504651658123484917078,

    889391565757005904380266469700600053531424828601721987974403995138,

    70352386070673378220293364466923965705219173652282041017788340104108,

    5034106362091260575809498994932119511804871604154207069391993561790694,

    328040333259686179449142742687497577715691495623439673279424332704571526,

    19583323127131046936936698632233812148424038134001306952012207987569990610

                                                                        n
      Theorem Number , 169,  Let , c[4, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[4, 13](19 n + 17), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4572006597, 39149016045892635, 10623563616616980201212,

    460929088757252577567900055, 6073871955267428405189466230522,

    33970798724066604519082116478937818, 98630909336541041420500643781635752351,

    169730620926820766054629527679944108071631,

    189966641828509813690869737690447218937241001,

    148026337874856057355321860253247350113379890114,

    84568560486687843887343896430574230518642627797086,

    36883570066268642191822213610276680356784247929193815,

    12683243887946782613080059721713838369342280694533620365,

    3530334918739103905035940659084787471419870155285158915902,

    812891882347522478133092998038250036683394254046588330099594,

    157688495957204180718613018667283308949659014719484143781627024,

    26172011659706280397171241988400939421651432039252791495510412457,

    3766211563835639335330408342529680609101507676630009670540012306596,

    475320337735293853208263648500268008453674263020357699654962488900665,

    53140827985988045457410022173203347092031772485020814633385177455888820,

    5309432863196028675853257669243417051837629547390060784372481024454514807,

    477772407498379550768198474472948310800136390888793120858004006855602782580

    , 38989591752560590479153265629042084796620320521239893901921478143414457\

    350620, 29034513290145320198494082749657266007554379517320822925120041721\
    45354110732654

                                                                        n
      Theorem Number , 170,  Let , c[4, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[4, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

37, 204738, 135585169, 32836437070, 4237314144453, 350531972141912,

    20707084780969536, 936568259173471534, 34036308998892441815,

    1029359042913839965960, 26602336411039041626154, 599722688951735807877692,

    11989106615029431008246494, 215386501946790266346578784,

    3515817606144360537287627044, 52627655765159452391106532386,

    728073325829112460653233049833, 9371660707021034913258353149962,

    112888991840155393793587890728313, 1279001462927351498447867088629016,

    13689849243463001838656411265700445, 138974427962684667557552570266588398,

    1342739295952263451223912579171646733,

    12385705269314509426427446037385987158

                                                                        n
      Theorem Number , 171,  Let , c[4, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[4, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

282, 840014, 433301474, 90592878283, 10573950233150, 812007701363105,

    45259074571492704, 1952843036567285170, 68239655089322021396,

    1996096845471007225131, 50121148644074680595810, 1101770662839007910468385,

    21538711286756480825672602, 379290412614885243926611197,

    6080747894019048581078642960, 89545755991977256694106667955,

    1220459558587193828527863283786, 15495789873142200735439373210439,

    184314555050880112760280769667288, 2063927919401457090547618960938757,

    21852157506821509509946812432670758, 219593361729967110993557116112162857,

    2101585304242992585519319728242433452,

    19213225080233461928845336773397743325

                                                                        n
      Theorem Number , 172,  Let , c[4, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[4, 14](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4344, 5278168035, 159332714609960, 887655925712402350, 1748754750137736628232,

    1666851085363043729324953, 920810201230926125798437110,

    330942420831414754842378659015, 83779343204945505800127622576040,

    15821981863254364389174452451934507, 2327347423172353056753527474452392656,

    275707384585285442203065020986754388092,

    27010769638386002780025567143313061588760,

    2235884867333236213063277472806992341517305,

    159170227384091370355447001254352765396101912,

    9889757983911414704215370735622975528136407409,

    543053668557981819846366592203919636041687170824,

    26635912162751864897568310683447617321312326147845,

    1177766437554424786344715709155403576876526239300960,

    47325724916421880375335673458504083989202128138731119,

    1740333151767436676305009002894789801246415745426042296,

    58932755661362270748732121949076460382720074090075553374,

    1847824647912415168647702513695124867986745857196110723400,

    53911141251444542944663779750766709129590361581111231982750

                                                                        n
      Theorem Number , 173,  Let , c[4, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[4, 15](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

43161, 507927931, 915013990883, 588820653717318, 191715121181026587,

    38013728489966990651, 5139991171308977670514, 510984232093722039200597,

    39388145530351316644956868, 2448416819752181449872872322,

    126471236352447661126403726581, 5557871590466489208849534510060,

    211757951081310960800302940754356, 7103800313910040734432382760524623,

    212533518519763062438720454407631929, 5732333908127860540149303982911118044,

    140665090034308510223749767931671213665,

    3165304528469466981422654622392804716513,

    65763885713768056428371867109110409045842,

    1269101839109586914668931891100194259035976,

    22867870938037223023141846288248618519339903,

    386542707544433052391461774501228384747936342,

    6154788426281374932094685577764926254407163869,

    92658598410896032201263204337908095032713117097

                                                                        n
      Theorem Number , 174,  Let , c[4, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[4, 16](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

376, 1624743, 1142017116, 313736651752, 46930954597464, 4533546138615901,

    313232016583402576, 16554216333727403046, 701454959990897443440,

    24668277898984564758874, 739152413016965919684864,

    19262364426997537000883352, 443820900205110286600698288,

    9163327109198070958359652986, 171424270033943654801377979088,

    2933046122744788104673143969920, 46263441840944059078514397177728,

    677309539965032848084880619821954, 9258252815982281001231123856295184,

    118768470382521929986799072415728472, 1436387883812542197075004052613953072,

    16443073201614922363978322579368030656,

    178807684024317210771943967083332727884,

    1852975954749489744608099004455809863184

                                                                        n
      Theorem Number , 175,  Let , c[4, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[4, 16](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

35418, 49053343680, 2060702790279955, 16182167394107165056,

    44478796476197600304906, 58318326131465129342332172,

    43698614724021555703018714213, 21028837200429117762961089626240,

    7044980600083909699835559611546620, 1742231475420077314126558519258620608,

    332415446659774384845277214458021296576,

    50641781397340094308014306417125071049856,

    6330587209014664646090036964476250446570630,

    663901770355259723249356480284794224431802752,

    59486874728937958462284873994584671699215252078,

    4624149103144507646768779318105021660709002003968,

    315903746660816032466487348226044868466578369305478,

    19177787371948090963122288391829829025214874176892160,

    1044519338130722932478891178343610073499900394546489022,

    51466683198698682538383716500962739886798932006704681600,

    2311008163898177998485270810246368709719800516158455083988,

    95179978123429010869005692669101474401705872064039143399232,

    3616173208868766754028885118968377724264825313232359684806220,

    127390598690936532859062541901451522182048498006870500263309568

                                                                        n
      Theorem Number , 176,  Let , c[4, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 4
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[4, 17](23 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

390942984, 1997006820275768701, 31051035128893625119530594,

    32070312349640824874370194412127, 6193076280122066438072963724910301557,

    370040560551716357560638497271952566419976,

    9155254833099486045655543145361840705291278077,

    113067478954501661762447881239063004923471777022641,

    792635475343962694896677623444438281337341519064504196,

    3461410272674201276327885487584178698114284910963002949295,

    10097578563975114542402490239355852678188996996988250237610998,

    20771209241096331775885082654879807206200505470489179391877623065,

    31450575813839981284775951072168385946660638882793827644465346667887,

    36291713283144094330996975715766836942206668192781081003285094990825529,

    32841258531479124839216711829404904125240570895460839670177559244798902577,
    238691950619213184104776083975146951796178633798030113127976073819815293\

    31572, 142177202691317861752451448891558747746044107913157915798729652377\

    38042165957734, 706137468123960608237933633909867130585961346645806086712\

    1928369892756122883436693, 2968104629087680080956764156422165477193740425\

    780973108369796504140410198439554323469, 10696022177040388464762800906496\

    86253033801028881798044253599592150986400406382002286797, 334230449330059\
    141045758370105688268369971499410950522436017389221664255728104152233116\

    716, 91474029754079744654526944345309116350862199787000108786384095694765\

    408495927213529694917831, 22122625277976339971967897252779499774589844260\

    044149639484670874093386026610247089375866087826, 47654816888266911641563\
    435808668581392874275769583779590209736833611358932765978839332350414972\
    73

                                                                        n
       Theorem Number , 177,  Let , c[5, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/5 c[5, 1](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

37, 1980, 44484, 631488, 6657156, 56679130, 409872472, 2602236008, 14844982148,

    77396342116, 373564139466, 1686096969752, 7174003056616, 28962893709984,

    111550676263488, 411738254436820, 1462039950688244, 5010924678653448,

    16623994216606300, 53516962223691944, 167546115838586393,

    511095862356420596, 1521741288228339172, 4429105855464558312

                                                                        n
       Theorem Number , 178,  Let , c[5, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



              Then , 1/11 c[5, 1](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

900, 469241, 57845696, 3401890620, 124643344232, 3260910480280, 66054211389096,

    1092175126730788, 15292213534764484, 186239193210364824,

    2013229934302071960, 19623844295137192516, 174661236781219134874,

    1434031574917747487672, 10952796492337571151768, 78370662556313582531000,

    528486835440223705984224, 3375853449416408472886624,

    20517120144152418182148696, 119096788719413090519940816,

    662517461270952311963626120, 3542419408434265833618019576,

    18253893007130080415933827776, 90863357383151588436009446538

                                                                        n
       Theorem Number , 179,  Let , c[5, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[5, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10, 294, 4160, 40058, 300470, 1883110, 10285056, 50330560, 225001152, 932078810,

    3616907350, 13259089402, 46229535040, 154151755776, 493835068950,

    1525746764480, 4561016308490, 13229097985050, 37319543014698,

    102611098883520, 275491344568320, 723415879076070, 1860660042199360,

    4693649267311814

                                                                        n
       Theorem Number , 180,  Let , c[5, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 2](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6, 1088, 48166, 1129866, 17981760, 218529792, 2170144410, 18387829632,

    136927008890, 915448058688, 5583600260416, 31456022385674, 165294806740992,

    816624784695616, 3817900354203418, 16983500940114870, 72214742661883450,

    294662383527965868, 1157692041888038208, 4392467094181490688,

    16135794844019459136, 57520912996798908522, 199384203139491115008,

    673235411934849186614

                                                                        n
       Theorem Number , 181,  Let , c[5, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

21, 2496, 94208, 2017216, 30198336, 350920851, 3366159680, 27737721024,

    201842757632, 1323389988864, 7937458791030, 44066975940608, 228589830292544,

    1116396025319616, 5165644085863488, 22764438618763741, 95972658599669760,

    388552222988099584, 1515620464038787648, 5712342387150191808,

    20855069626938047040, 73917304894377543104, 254843507536465038912,

    856167057075084730368

                                                                        n
       Theorem Number , 182,  Let , c[5, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

64, 5494, 180282, 3551274, 50209766, 559247286, 5189881856, 41635275946,

    296301041370, 1906335508544, 11248884574790, 61566659330112,

    315361838775168, 1522920421416960, 6975551179869306, 30459156988956992,

    127340441328150182, 511597310025498624, 1981489005225942336,

    7419374545029867072, 26922708552979330278, 94882580278643250442,

    325392058789902379584, 1087747366878088424106

                                                                        n
       Theorem Number , 183,  Let , c[5, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[5, 3](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

73, 47424, 6081296, 357756464, 12941145152, 332554108592, 6603710110672,

    106977036052536, 1467615255186704, 17519790234423200, 185740399255822576,

    1776726024792601952, 15528632099642367600, 125276711747721415696,

    940758552863578917663, 6622221216054257565312, 43956633165495782243376,

    276531231317007200080048, 1656017759237339128024240,

    9476371212489167338579936, 51990814861036440810954272,

    274283391133504724698471256, 1395076487166274687206476752,

    6857053211814783721831066912

                                                                        n
       Theorem Number , 184,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

42, 11310, 815962, 30514806, 752111528, 13787699398, 201891334016,

    2473419061344, 26182846128946, 245169284651718, 2067041757906944,

    15909681622907240, 113026119054324736, 747791785588277248,

    4641626275776564578, 27197496859061900770, 151226993209113238186,

    801525025733736837806, 4065124357141698185888, 19795423339001054120832,

    92827367930920618798326, 420284943239902862803514,

    1841526906820151746235198, 7824942710432732978104354

                                                                        n
       Theorem Number , 185,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

154, 28616, 1755998, 59595918, 1374053394, 23969726976, 337623430144,

    4008187906478, 41332531953048, 378508182381760, 3130469035301342,

    23692622140571776, 165829322560910124, 1082637511108129410,

    6639973828457146646, 38486034195950966296, 211882264006360009970,

    1112830904485648781818, 5596825448806621830528, 27043261370950474414510,

    125902706618872759519776, 566211968982572948098406,

    2465346742061794598146048, 10413934231225326958023562

                                                                        n
       Theorem Number , 186,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[5, 4](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7, 20440, 5433696, 537222520, 29513432568, 1083908446760, 29523237109320,

    636679080267839, 11364058302076600, 173304073398371208, 2312309726246126592,

    27490024425679261640, 295432677947656952512, 2903660255101212989920,

    26349969099773559031743, 222541792673859853850328,

    1760961958615567570104320, 13130358569363150222462816,

    92710295553634640153393080, 622529095189585652190521832,

    3990207737388812488729235944, 24494376625464924196510537150,

    144424522100040123529332994752, 820068736469039325422355000320

                                                                        n
       Theorem Number , 187,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[5, 4](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30, 49638, 10979322, 981466710, 50464689996, 1766727900960, 46385674809344,

    971364098433546, 16923301528979840, 252873696543307420, 3315447339840403270,

    38820089339647243264, 411635106750784101270, 3997732463433301307520,

    35891638875281985430494, 300203530885644902002510,

    2354635645394418890258720, 17415843297074457439839940,

    122059353683036130841891840, 813997542103897417453033700,

    5184354823593436641420161730, 31636753111972148144177022080,

    185507831933912305964229986886, 1047897832291278397946910984280

                                                                        n
       Theorem Number , 188,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[5, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

110, 116308, 21796290, 1773398322, 85624043882, 2862991361770, 72545880655360,

    1476458398504960, 25123560071381888, 367996280626362390,

    4742838448897961890, 54709292171642468556, 572517875524097741290,

    5495236551170820494142, 48817960910359209671176, 404437120701837820070290,

    3144690163541638605955204, 23074667376717595682524830,

    160536667514072829181437410, 1063356968835117558726381568,

    6730000881057991879524112110, 40828557222605697746351954952,

    238096417134144456006982979270, 1338065196688886010105687147520

                                                                        n
       Theorem Number , 189,  Let , c[5, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[5, 4](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1084, 582830, 81883520, 5614637632, 241159592960, 7394311062324,

    175120917608370, 3374457176541462, 54869852000659070, 773312507934378150,

    9640677525047775502, 108019280863652312990, 1101694080671920360190,

    10334459670102371994112, 89930504727766256799840, 731213829808678485153280,

    5589244793166237841503110, 40374644659823230282632082,

    276872829199361984895334026, 1809613678182235153070735360,

    11311877862576518642004150748, 67836224584936518421024565210,

    391339873555185283861636075520, 2177076957202139326216690329600

                                                                        n
       Theorem Number , 190,  Let , c[5, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[5, 5](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20, 1304, 35540, 607640, 7657040, 77381680, 660011368, 4914276480, 32709263160,

    198047776200, 1105442938580, 5747729305648, 28072155015480, 129668271427040,

    569669974927720, 2391694905496320, 9634589288163404, 37368893249726280,

    139971802954625360, 507648109997010120, 1786810122318535840,

    6116092364459822816, 20395721145923863760, 66371259792080177280

                                                                        n
       Theorem Number , 191,  Let , c[5, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 6](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

309, 107558, 11041519, 587535506, 20325409959, 514981197111, 10271201573338,

    169142686912055, 2378273480549203, 29266428595986947, 321177089356981345,

    3189930753581558288, 29013523094035647204, 243994196169177586871,

    1912405732075142929514, 14064023542135827029532, 97597506684942283415796,

    642226507450901422098213, 4024342931417911301132894,

    24102533890152350134463047, 138420857504462415754749563,

    764468144964376652067076724, 4070493225953612831276334109,

    20943939191028941306840577377

                                                                        n
       Theorem Number , 192,  Let , c[5, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 7](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

17, 20428, 3924687, 322930052, 15729769384, 530644769848, 13575589685459,

    279177483925852, 4803780204414936, 71199615067544108, 929078347848027636,

    10855852771851501404, 115121448718008416226, 1120120619062841914160,

    10090015726600631698349, 84781119343516770126056, 668723739550769969107295,

    4978462411455143543441660, 35146546218854310266432438,

    236256219089806408105012160, 1517585346171510576211790023,

    9344717832057438348582909940, 55314988309412673886573725699,

    315553458786591529999778081176

                                                                        n
       Theorem Number , 193,  Let , c[5, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 7](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

92, 64345, 10002364, 727865318, 32625628612, 1035251489395, 25253660780876,

    499752747771186, 8329248859279652, 120159136550434963, 1531813775033118932,

    17537452111347801949, 182655688151266743628, 1748864380947907397798,

    15527304707080245716632, 128767036521574023512033,

    1003586760311902245772420, 7389879395860431932578283,

    51645565298701274074140532, 343929393366284192225326981,

    2190096143093642126763773704, 13376910085724405461893659519,

    78585019303692322074055087036, 445124251607088596993317720476

                                                                        n
       Theorem Number , 194,  Let , c[5, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 7](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

418, 191896, 24756263, 1609670772, 66752323697, 1998743029068, 46588399691881,

    888505993393300, 14359176942196778, 201787425705521736, 2514759848456122394,

    28224651869703675876, 288835719627842323001, 2722313160176867890584,

    23829590184681068696189, 195089157414720674413120,

    1502717615896143564071574, 10946437608574494532886688,

    75743559176605158561648282, 499779229489494766992319016,

    3155365400787583938089769081, 19119189578319486949120247440,

    111481640303978670912503549042, 627038898362768309977330797356

                                                                        n
       Theorem Number , 195,  Let , c[5, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[5, 7](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1509495, 51856759832, 192212595296610, 215780200034919664,

    111090661395323970385, 32608122824429526971560, 6223616004457697264019942,

    842344670420631587216836040, 85918930487447209976444974068,

    6906162234118524878976320458560, 452563071745360140647874506625245,

    24827939153375509152666281266811924, 1164755340870317968332836066516015700,

    47540747887565926640038143385746272616,

    1712557772926119695230129705992202510930,

    55103877838653120715316033776745675616600,

    1599921016846354277800922043931901781921490,

    42284959896385432146298762734044160773891360,

    1025002638300277797908520865335605105441092177,

    22939146131056363226198193988500910059004593280,

    476712587605844074415713550675020661709842444760,

    9246681931628382447753710075428260295457285292192,

    168167514902466310671526843278795489567114385890200,

    2879339256896294249773949075413347862831004817577320

                                                                        n
       Theorem Number , 196,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[5, 8](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

33, 3900, 174858, 4642521, 87292755, 1277508342, 15410720523, 159179522331,

    1446158922945, 11787180773412, 87505505212245, 598748715657675,

    3812160367741074, 22761092345003304, 128266811160919323, 685951287637782045,

    3497332456984890183, 17067510454652961864, 79999799076759336762,

    361243887117262505136, 1575635831479882579308, 6653870894682924474723,

    27262415157854112560973, 108577326263673148161465

                                                                        n
       Theorem Number , 197,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9, 9239327, 91785911200, 163295104270275, 107570246603196718,

    36207260940961257891, 7435204133960212516629, 1040097148040502354621505,

    106752164325519222167925717, 8475289189376441277926008170,

    541314249186515774195779635287, 28663780823956645075484884858558,

    1288540639245805967847042223426224, 50121117748459878053857336223945725,

    1713477398645124477923581185646012263,

    52155975094873412406636601037595947076,

    1429029128579740261547561357997465229006,

    35573667331755823780134339600067426091016,

    811036701166536598187428781086824237334881,

    17052587749223799083692714225252024893130797,

    332669326145947076472427751959076794422111415,

    6053792185554029658365934534421385204099753769,

    103249870323685323146683872968388460940146697207,

    1657408890060398488240725413212455443965037364591

                                                                        n
       Theorem Number , 198,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

52, 23807115, 191475901392, 304960922309841, 187077623901213285,

    59843736470865831471, 11821751059161274037433, 1603533791100438474639948,

    160472665100109518694011369, 12472570959348383028454489567,

    782273218105314273486620832633, 40774310906277419387665042062599,

    1807681770666550512065492617523915, 69452378278931829522567197720130768,

    2348238535551558753189219817964182830,

    70766892538640927940450994419122927825,

    1921424339923630096849392812606619914961,

    47435597050312440991670695606540819206424,

    1073251173850590651103602707766501306347609,

    22407345033402260893031297200910035876619520,

    434285335276954419133824417576809180277720688,

    7855056081742341040774277937708977598719050579,

    133212761964407911482902417451069453841099805679,

    2127046764556426103704091812401296802262267025381

                                                                        n
       Theorem Number , 199,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

251, 59676301, 394406978985, 565168898022452, 323630841829458727,

    98521060122889774128, 18738337640994020903181, 2466020345719439601524729,

    240725469160676550791974716, 18322728507785356698906595857,

    1128768863996968569146549921129, 57924216809794414569236877316836,

    2532987338496278044313053911969793, 96138090095309019706359932369358609,

    3215094022400789563896442596308711064,

    95936155821312962955235947976224316702,

    2581454343135809292564225275410985179939,

    63207311473738035439984729182664206585090,

    1419299317994604124088492278911111784009630,

    29425492698744885156271714831987334335024703,

    566617163888186309183211647961479434000568826,

    10186852131621917542011856939155601264422479068,

    171785315390249981025396982656953796374233316512,

    2728485809798743231730227612353670001167972446540

                                                                        n
       Theorem Number , 200,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4077, 348411366, 1615121674694, 1899001754057550, 953817942814060959,

    263981503004426355468, 46657119534015597313573, 5789650238105583860474850,

    538405436849204582806527216, 39336793725180456820598691476,

    2339612654022303608646858154113, 116438051396989954606348592832253,

    4956079512963011089307481696084363, 183635273527192182174186099825334629,

    6010000981852937107143613132673634974,

    175864356740263839417625142271563304513,

    4648732618054766025713180325687777430080,

    111986151891176451838343699420892564714469,

    2477209669977718919554862668826162033111462,

    50652091384395549605850009242385314211796929,

    962901205616375308579677954730041846536386351,

    17105427180708882568243976559842904503795349279,

    285248391751783132579942582170808507790944473516,

    4483399799221749977483624181472300852636055454916

                                                                        n
       Theorem Number , 201,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

14417, 814627733, 3214685665476, 3444807062132842, 1625464008631544522,

    429718863130877067084, 73300507799107064351701, 8839463363610170024984945,

    802791539091260324891769627, 57490248168457171126485224815,

    3360875982440243645748836283230, 164766567097961404637693913173025,

    6920588767590004786416036388299467, 253406133066779840080109984601160360,

    8205618245624174277611626553560026756,

    237808525126793833725258968652835846560,

    6231160005898769958399414606148447931242,

    148902858908016522682832604897650288322597,

    3269512365343316789386926390520991514714265,

    66395898814556438298456847916109143264686410,

    1254187125079643461504783098025614831671038554,

    22148277250504242160331761507221686840703511994,

    367301488926375656357790931822774425911422097586,

    5743159014725717114299985058357655835523350455143

                                                                        n
       Theorem Number , 202,  Let , c[5, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[5, 8](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

149071, 4202923779, 12351536292898, 11114581014499290, 4654775578541716872,

    1126587478010598568245, 179390202438618484945662,

    20462172564083469338560554, 1774408389021313605828998975,

    122184458776039840694051267811, 6905367053381055753692665724907,

    328671802200863530665614986097528, 13448772679603025319382954659324057,

    481085156262227703278110270368145893,

    15254413275109540372817855317866563543,

    433756100745795035045890565724760327382,

    11169900468506536365556984499190776265588,

    262707407020411070943894256662448877350894,

    5684405731167142620197529646213018170532675,

    113881163650729953989031583319833583336158419,

    2124225162529059998091390002293312788525616485,

    37074650883543099724133580530438058382414359476,

    608120531079148927244199655677630454608351264327,

    9411181498881116388519921174976064950699009965575

                                                                        n
       Theorem Number , 203,  Let , c[5, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

140, 165414, 39411012, 4163030012, 261157014772, 11290005478001,

    367433421687626, 9537800173307146, 205590828628416104, 3790031915523720313,

    61103838676064685482, 876698313059656956642, 11350928905685291236964,

    134131440425638736602531, 1460210585029742914270308,

    14760360177011030821510474, 139466260400501272649522136,

    1238834985424075666209157258, 10396252256904431467238400038,

    82781230554548635257071614076, 627803863230370851788709185912,

    4549965543248739913682581519862, 31606652189884636374504800028842,

    211004920146111318141345088044946

                                                                        n
       Theorem Number , 204,  Let , c[5, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[5, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

715, 540398, 105470126, 9871758048, 569515612939, 23126489262794,

    716545975465198, 17871129008683064, 372578073024932672, 6676006666103260162,

    105018059969104138148, 1474629434101849984720, 18731165768603879727087,

    217590096018044232836728, 2332538143297311600329850,

    23250449220521530930075088, 216895807940915289338389916,

    1904132052332356155378527068, 15807268153460671214714784318,

    124610386423755275222409977144, 936254424411705745770721589304,

    6726612526336949678082609752026, 46347568647687034794375087586282,

    307056324563751747951020067448504

                                                                        n
       Theorem Number , 205,  Let , c[5, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[5, 9](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1919684, 219044466515, 1943377177432736, 4499471023742266092,

    4365741886802035091140, 2269446675349117402417518,

    732358481666290828991741180, 161619191305608908834600349856,

    26098818802316047136086261424012, 3240691988725754227763782416476725,

    321279694965594083556125998356058462,

    26188534325171566817778542895810848220,

    1796827497200908299046716029367910250152,

    105768873186352001622965100717537540764774,

    5426507338113960567805256648934174282139660,

    245889796881758109188424791416004130712732812,

    9951602890484466701384651097823105081747548060,

    363208658823505428648160908188701564504090873768,

    12054351216106186710398258531695186353902547552288,

    366443540094524772221768579290490766334439842297800,

    10268650330947812157502829246116752255093519826070472,

    266752893870148037397263309601209321717524276776840415,

    6456092107596686094961971482589186775844048988305685356,

    146231092851063720758012631160553463126080905880585293586

                                                                        n
       Theorem Number , 206,  Let , c[5, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 5
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



             Then , 1/19 c[5, 9](19 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

182941519, 364567433815912, 27636331570816878460, 384265943428879714281660,

    1795566906974103058887326335, 3849634502449769611037288301200,

    4559859063042440330695803037160768, 3369595864585625102630840993593443660,

    1690853087510636683020776519833971245270,

    612960510875796151277934963055681307496360,

    168240387340150285234185383600423878908711924,

    36263913408761696527643509175162060550441234808,

    6320125510765696257864163943862793561669279819345,

    911956525482250073272819169832708007434092049980880,

    111101749535054564762648507363882746303654719468651684,

    11616781445605942495533319292103007720333768579759363122,

    1057069319961552475353178006732209698368732738113554447804,

    84711441648476063212207165744429834763810130734958652018760,

    6040440981497267427397973837537471087789614373626572331858080,

    386701330772059045697563187250502186102911546610472228668823544,

    22401635951445416754169933699070472211437051048981407834502228659,

    1182495296910870922453259284164558274839422576970777686396468828800,

    57229170757561043868196792391789146198116330302106684139848822854520,

    2553470914560790854081527281089181226327780882630378372425458613399800

                                                                        n
      Theorem Number , 207,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 10](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

650, 6161285, 6568367265, 2390527133449, 441160390314605, 50309952612997915,

    3982601129376393595, 235994741187845855460, 11030380053959352363756,

    422457726294771425690120, 13645448268217282924394845,

    380131876279369194620455940, 9297647171662546376144048350,

    202581118761416761945821677357, 3979322087892210178018805995050,

    71179438909291804616942904956220, 1169290396873618933602186062149780,

    17769280251520191437495898564162785, 251377083208836917491552713570613578,

    3328646969742623028518397713670810290,

    41455602405883306160834277296004438715,

    487659182024940724342754885787980066870,

    5438842928758770947724718092975407325990,

    57705670446331392162465199052907223382191

                                                                        n
      Theorem Number , 208,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10, 210842373, 12906151446605, 97665541287304655, 223075813655089695365,

    227477237235646152294680, 128318928205740105452778122,

    45730550431684391485657167405, 11257279893225571287755547943850,

    2039452363473303930442152341889505, 285015807809950727319211738714466090,

    31852844185824591720609577641013478984,

    2928607730870107433390232819294450098610,

    226621801682757445519590231155552089503975,

    15037145784470386551875016114122340528253740,

    868916789042561012551796112573685517366206080,

    44299722726403150325204423525704480440739638144,

    2014896672876053267055505453622948801452130506510,

    82542274752270613642134974805512789723421986231070,

    3070895533509934763841666936157748546659465781587510,

    104510317009280532153633104083640094746080093658007820,

    3274334502349874440768515345444517408707072051345521746,

    94974068219155198987885356302095464987996654240058517655,

    2563221335344007246342951311732746568299498326735407494305

                                                                        n
      Theorem Number , 209,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

350, 1414331020, 56556908346427, 342527513211716345, 677919749944117714545,

    624050431912243152140840, 325638991778064684520190430,

    109082140871532102586211672423, 25522184139899850404905663680890,

    4430647263052621275216784232425120, 596988783841401873765823756970341120,

    64635708359640691676837245568148455890,

    5779279288080743650057628950005187117313,

    436267019772374981595180798359584009768565,

    28311926363281366703208134165002194680095110,

    1603501496561610298733800988850904603502020870,

    80273246484808765981007229480027107556927920745,

    3590699466405554816538474114689417083235487937668,

    144858843641711552279591563643600445544795912446165,

    5313570198568170559341930729546537445666555181982385,

    178476661148287828559219125179985557375544079952608765,

    5523852782904784993467280265775711661782240458757431205,

    158406403770210723013679364046457014116775848173648230515,

    4229770947485053106504023549432955546390909840470196807895

                                                                        n
      Theorem Number , 210,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6945, 8659869250, 237547902477095, 1170678097015767034, 2023725422815522394675,

    1689365610655312018307105, 817819927460117801661457040,

    258006917194207352345762909225, 57457521430610119215599069183150,

    9568073981587795389420868457626115, 1244001888170190492884192256551518570,

    130567230206480263855705420894074103510,

    11359239494858115498148537847089221759515,

    836857402751232334275167982785062069386159,

    53134447048344806760220027485704593752582095,

    2950485604550865466073607373982280780290515780,

    145072355064991836862684501870386718868128370910,

    6383296226596320401858908696558473972775533753960,

    253650860769478853594255420419864536107197763053345,

    9174940642381388100858490033700855624167157234833550,

    304202823400866947428616566248456137592116645593336240,

    9302025041095429255279971597580049383503905096990092280,

    263759096158882706801596651225099981543110349430997277450,

    6968832253773488926405867355476768630772378103033768321747

                                                                        n
      Theorem Number , 211,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

331885, 113565690220, 1903086520852673, 7071509081667637545,

    10113068577757844413835, 7347549067501613882366455,

    3194384284756341870491995485, 924274818522694106776199379308,

    191616978818124710008139690059490, 30034743975083565059721922539457225,

    3706823063839428237792002177545598860,

    371774198973387506416515634979200598465,

    31072284086486134241327414874642351051949,

    2208749835792893750758667808319343883862955,

    135804202419780820773476914050586003453940895,

    7324765563098557081741165860703742457070990290,

    350729302952070451587747689413446988193861153905,

    15062092783003927004100302279249017070024899954063,

    585284139147446693914735550007035844384670388060390,

    20737441359538095233133190974456900172047197993649100,

    674497171273176058543033597038281816971931647166087460,

    20259518563683051424330975601605366078314171054038434430,

    564938436371128275792481219638534847637254839955011024389,

    14694332970987251571869675301949863153683148342360253797395

                                                                        n
      Theorem Number , 212,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1073341, 258935628115, 3741445524157820, 12734023040418154055,

    17151515871775618970365, 11920604626897891190000318,

    5006425400125986510231410650, 1408648769137566294423330296065,

    285333206550128196754846935171430, 43852011066591620344186394866008025,

    5320973623270985162244059625383189936,

    525799114773934407996679339672372951730,

    43372277066321010103532299215042225442655,

    3047167511731134690499883604373377715260305,

    185388658193129900264521056390135814262981280,

    9904055920094506342405300696839415852618655686,

    470117669990835980371432704278951184698774021305,

    20028560716752624126806339251040911642621419294545,

    772563608483046303348884918156244702976946383011020,

    27187211088927213972734784467384352785267299134699060,

    878702865667906799235194708131859969051741828168406714,

    26237951967148859308147290776334570330203603979792459585,

    727624026199967234797870476934835220917875157122615952340,

    18828241348331848306608092992366689574211962227198976499045

                                                                        n
      Theorem Number , 213,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3317615, 581481767895, 7295204970067960, 22807799039477391495,

    28976285538335406846960, 19282820290874462363995310,

    7827902209520440411329569160, 2142711893480420865087049381950,

    424193244906421683445158443419526, 63936611382987713107783447218299470,

    7628761172943546288633479141902815715,

    742843460303092828411534162629969923455,

    60483863907056166691051219466779536648865,

    4200266918438950855142072218289036900548063,

    252882918599833415366323054640646033827634680,

    13382246632276367452584690546899549674977012635,

    629743327613912398531786174779074621296211096500,

    26616995102893265022601621969558661285653877304875,

    1019215717259696309871357600818201078649878607034900,

    35625064150344876307121001451794686731339697827816260,

    1144196470262513812645492338131454074795946971882697210,

    33965722580891344611442755051022767555321607674487818160,

    936774553454484803103215970601157203303850678922019466725,

    24115850932420228416927141656247686234013208033095772588208

                                                                        n
      Theorem Number , 214,  Let , c[5, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[5, 10](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9851440, 1287206918011, 14111904777263295, 40637813259060999575,

    48768677322640837649830, 31101630869788803975014780,

    12211145457086427781415896125, 3253089751739595209428147692115,

    629617906008871733478100187311420, 93091711610992232198146193741684710,

    10924358876751554233668140041779802775,

    1048373265358698809485537751133824832784,

    84267216138135814687302829914473409968290,

    5784837641550710437515914169470966368609760,

    344686847122099670163247598434057512670783520,

    18069398646579671725540824437624894917649325735,

    843033607567494152431166554745036184107113172158,

    35352063163854210314946510193381771899912214739440,

    1343890080110710696591006824497760307254420209574915,

    46658351409463626761205600206042475538669997990252435,

    1489213303351955593430066756482785181423005357468224025,

    43950403911096757544981628377561508344074680708364304780,

    1205552583372784753622798595040630563569888973371579004020,

    30876578160050215480142430283627907223195582595208820518200

                                                                        n
      Theorem Number , 215,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[5, 11](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

49, 9360, 630737, 24074064, 630702335, 12559275792, 202297236400, 2747178852768,

    32387185755011, 338726606234400, 3195251389001154, 27541672862430960,

    219201728275520690, 1624723793156281104, 11294905527161423300,

    74088546986092330656, 460890159743505946916, 2731019581389867478704,

    15473364368814422923238, 84104981704415151396000, 439853404781813893890278,

    2219070463612184536183872, 10824710583730532188271415,

    51161612460249261551507568

                                                                        n
      Theorem Number , 216,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 11](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1141, 1340640, 378421401, 49067086960, 3806311410412, 203235963740640,

    8135157583361018, 258371935191840960, 6776943316296853980,

    151205678931179206080, 2935286769001350357510, 50462989022649090837600,

    779302693632749118060779, 10936857780035844003169200,

    140837071227077816422172586, 1677641021256892495180118560,

    18613715225058296584335555876, 193505048767638680375205305280,

    1894557470063929894696324406415, 17548062628482360078531714034560,

    154373340021968123766755113510722, 1294365125423797064384974862900160,

    10376152028068100137885712905817393, 79748579360118133885558102197490800

                                                                        n
      Theorem Number , 217,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 11](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

21, 957600, 2038323878, 1177362365472, 310050192637291, 47444605603523984,

    4840673797354895700, 359299603312581902400, 20591441508372925402946,

    951030198690936229793088, 36551422973388181176640595,

    1198315015183494639414370400, 34175102127030196349357193411,

    861396162312689033076210365568, 19440560556797137174197877477795,

    397143955954973445271668331495200, 7411537162905785812775509218440995,

    127348996138411336163544205369925952, 2028365599074809964190313632622232117,

    30124092985033193421652372483118126736,

    419310818451421501011825727296916297761,

    5495232108208044517802844220949712609600,

    68079353511435820461440473836437704406886,

    800179318563219867087719941832019223562880

                                                                        n
      Theorem Number , 218,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 11](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

144, 3217108, 5382546768, 2718793864580, 655010200135200, 93943597832366010,

    9117647269868182032, 650147955927786557723, 36044992159035064884000,

    1618777335809177870275427, 60734541186945528372844608,

    1949744951628704028101500824, 54583813411890966310196493648,

    1353255335759949924783088861725, 30090862928039266229396817504672,

    606503628129897904649815915832440, 11180790618400196007788620130008704,

    189969773345520061420418690455795992, 2994652776179953587575124018368748000,

    44051742806600314151440676014769344745,

    607758537372292215787242112818960795984,

    7899323052425990737250208041740105794644,

    97109614446428993606592172441362418728000,

    1133148118075393328522962672185566307654825

                                                                        n
      Theorem Number , 219,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

815, 10317456, 13884740913, 6186197221360, 1369393452429066, 184551382279886400,

    17066523648815655379, 1170436963452800348016, 62826535597467866975880,

    2745308045987611146824736, 100597908019341297444408990,

    3163543340400688093937524800, 86964683147220593190028277047,

    2121258249341326321936281973296, 46482677567011148354566905045510,

    924546518159855045989267759214400, 16838929910275780785627129862745943,

    282950075111451878957470267255535376, 4415033991351537860119371967769072561,

    64334554356944293240948895528449540848,

    879827778704136860756303034895904323575,

    11342281296178411477183416311606982294864,

    138371642512165145282435353867516109837415,

    1603069114310766955983353745403944109049328

                                                                        n
      Theorem Number , 220,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 11](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

17722, 94197408, 86698815600, 30733211564208, 5810826845257870,

    696310197066977616, 58728381392064894706, 3737587440956825858880,

    188508602049348811667262, 7812041271454174287978000,

    273441648461722907928353316, 8260706355790482904307830704,

    219151655629658019138146698406, 5178324972265829625579782137488,

    110266671444262945547682223936230, 2136930313122808449016416429074256,

    38007304654296820740882302359580120, 624885056124652997570829733264781424,

    9556483146499449878813195161980879237,

    136686817910508426501365472669184296000,

    1837243245730903386400029625529335053573,

    23305601165665197547102140553652045402400,

    280057861028035720551572501796661791759808,

    3198873611931345974159307773480278397966400

                                                                        n
      Theorem Number , 221,  Let , c[5, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 11](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

38787168, 127667726380061, 13074449468133940704, 229206674597867387664935,

    1302635493106165685738980800, 3320841206376682885511004967929,

    4603854960517727767912961230003584, 3935022472088008773628962713756832627,

    2262742632791547742624700561567528721600,

    932903928154277008850613205456265346698482,

    289379376421580183135755411828976002219962432,

    70114532757566959859804049780912098381243952405,

    13671959720370475498774615647313239686080634505520,

    2198231809330589796238651469040967765240500557683890,

    297328422429721629056800927271309330783336325936811200,

    34403536914548377178633088463041639495655053185218773590,

    3454176384843754922957588587260675130638806874811604923392,

    304609984898074473062125673325124640927104326577483046338608,

    23843356041148531185936762303247797993588737828258783585436000,

    1671828040315883445459021854773165106429575154540259383120040799,

    105853989000860992892586613505118700290262644087590487310777567680,

    6095334663535466471890507197991432888457725445839729606388946633304,

    321219910560327750338891550484247468603573667491061560383349014728000,

    15580083654475917468738257197761080265878648573725215805421252473421580

                                                                        n
      Theorem Number , 222,  Let , c[5, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 12](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

33, 141185, 74915531, 14875754144, 1599963053571, 111710721118534,

    5625921832795651, 218736609245168309, 6881386904171587115,

    181248390848367144494, 4101032233460397411285, 81323905382238073633468,

    1436023294385878510494807, 22873556421902309256825449,

    332173659955146733070330385, 4437429405776896324957484605,

    54943436627025123202611631239, 634643047181572317332138269442,

    6877048783536639839048045540091, 70250686980169891022823724694086,

    679412073332226149906587823771669, 6244421383180484794062175951095071,

    54725156650252701906175312022085205, 458693070786503451154167070579136453

                                                                        n
      Theorem Number , 223,  Let , c[5, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 12](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

238, 552871, 229730911, 39526753444, 3855948668523, 250465634735334,

    11922684104273269, 442885274651984580, 13414268414527447817,

    342111749044742323760, 7528328131839170011147, 145693951419367537242980,

    2517775360204218335402944, 39338477650337185877117141,

    561444634053679090632703774, 7382964337785075873338364140,

    90109268553332867364184399572, 1027191620273907451752493326891,

    10996103888801006954370068736058, 111069023123776203577432582270493,

    1062981737988951592575869987260277, 9674766047345079406184286889434626,

    84016297667139268327927443106010738, 698186228271006784507056863285505694

                                                                        n
      Theorem Number , 224,  Let , c[5, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 12](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1412, 2033251, 680893440, 102713928174, 9144710398937, 554683202648161,

    25019072198176586, 889469665384055440, 25970571325914336623,

    641957500709659235125, 13749206223028374333269, 259837595679894285783240,

    4396675513186532520791539, 67411071519222647458355642,

    945862155955346817824379427, 12247180355469734832817866308,

    147379814326314257769266398584, 1658374379890444417747486139297,

    17541512093747618234209769977277, 175226240498776368891977078185869,

    1659763392426699361409406989716439, 14961445550072051743399989749088598,

    128758486676078164154835595445587467, 1060967332662256546421310855834270150

                                                                        n
      Theorem Number , 225,  Let , c[5, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[5, 12](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

924205, 118833298993, 1194328182443243, 3127903138259812325,

    3421967332654168186885, 1998488869630242054905245,

    721991793457792037293139836, 177777278257360432539856734250,

    31932130445531768646738056679130, 4397620930611399730083766767924830,

    482257878482843884736959479924668250,

    43376216336271709386650704245032751270,

    3276397564789242141742017497430288055184,

    211871308522487002772507794812890080888745,

    11917768408463511672046478455068654259437875,

    590972297644820888048340524056228746414452770,

    26128443561684940527832433874749837844024909657,

    1040058003353445174215453592116058734923938986113,

    37588408271132992511724489010364196407277964049895,

    1242484242685340277167852751488595356192793497821270,

    37806784209396775905991469839483224536044799038198035,

    1065044631839390744328411470673028422504955634604452560,

    27918286059066118571422398472858085315531004471566292085,

    684074235973930173046061789766052248056329569833660332465

                                                                        n
      Theorem Number , 226,  Let , c[5, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 13](19 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

102434440, 4132281314308127, 2416814271736006971840,

    175258681932260579354039936, 3408616502099088652497178166912,

    26149406892672233700104543695510280, 99240431727608407031668368856464783872,

    215755502533278838609947728416332959664432,

    297434122271535793385366698829550921934780104,

    279923739986399998946083834747606405996118645309,

    190150739500905812038662128946609853537569118350080,

    97354456178204134568045552372705701075654323212081920,

    38882955599748891537176738643625559866440939422547451150,

    12457528665557173551322012694209071824610621997180276859888,

    3276205374919760143348450504185180277061769046883181651783168,

    721004031982593915612255751501309587581272051841815473531537616,

    134961210267649889982199017808373421368551423490397531882324115080,

    21789114038014190097336613953521959273470742245648137835943581433706,

    3070843385513120724398532040679231937513131762031151733999778371412992,

    381781733541733801018282860528287986461654466266863601345621167220706432,

    42257561226714796905670235349928882917391361827493722769326864579430585048,
    419807797110078206472607141024900921969804575036457738101421079056048427\

    6984, 3770372855143539238392819383283624191120697494980440700344161677588\

    67625568000, 308107252646621314902882382065247963244421597824147230525292\
    16747189358454013024

                                                                        n
      Theorem Number , 227,  Let , c[5, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[5, 13](23 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

172167, 1558229664804352, 13065886694767929992960, 6255977596305592882519237888,

    550361271169142657134727877328168, 15208074775829973220913518142380824576,

    178232806440534692764739427822188097114096,

    1069831921308228396450326512286592089890380216,

    3737499716264455352556219363992260392841276979801,

    8327325420247683075441851778447715422324940789912576,

    12667313147722934849740210951512381962869329842985363840,

    13863755413993880478506840889187769732490801241010062252984,

    11377687085860084464544974178219399826574473024881791191812448,

    7238906066589366908865452001751219337130317343392893178388088832,

    3669448782652745922495979758324322110739560216122848229515218224112,

    1516036462379417518952219989342616212088041931548830299196023735251886,

    520379411175297413799427203235806300778842603196007975001889525656792678,

    150842496700530874830406114824923878126334510146401311720855620228954759936

    , 37447558918229574444726459408372248027184234329471254259704902976790235\

    926400, 80596710857913551586579873792771037826839899594625338008991509178\

    62147486084528, 151998528326460680310032365582106131946765685289654068367\

    7952802743720943631948840, 2535546070764814053640701371044789856694234738\

    58364161663356186114531965751388659712, 377251758298007271075918862341025\

    29283136317334515629649757891831475782711465089931216, 504362665055614697\
    1769325673121882711476925624338838419083320099923495476011529168969872

                                                                        n
      Theorem Number , 228,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[5, 14](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

68, 19475, 1857815, 96829909, 3376299280, 87742490751, 1815675641704,

    31267784471925, 462350660024381, 6007878759057410, 69831589694429367,

    736253195738804634, 7120714017325144005, 63758039616162325280,

    532577810311039732125, 4177022833050841880601, 30929420037236481392777,

    217245708741474280268080, 1453410054450943379055124,

    9294863300893939439085345, 57002093442818569958245796,

    336163786286640235476012981, 1911220630506546782994385640,

    10498936793347503402417424976

                                                                        n
      Theorem Number , 229,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

323, 1114689, 643354630, 147866765779, 18760670683155, 1553615990665670,

    92819512207563348, 4272428410395086403, 158642994413555713327,

    4914970701225981678374, 130347425244884568160848, 3019002168041790047781381,

    62052612280080390711631966, 1146732378303928069796631384,

    19260378768063981240402728581, 296694934791118835383185430194,

    4224218330729756710253654211900, 55956056246524413901412159662918,

    693586952857339291883134712247566, 8085004781883217520763249512275779,

    89021596506751977637297234376818722, 929472674688241644167410531455298640,

    9234400486135019300064109633825766646,

    87570549133036235588743050403143536457

                                                                        n
      Theorem Number , 230,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2091, 4395707, 2025779568, 405740455166, 46774037741481, 3604727255434446,

    203503197762594598, 8944503956969846260, 319546686186623839275,

    9579230920081409677096, 246897124627642963487263, 5576917980536363663451207,

    112107166750648041281881582, 2030891575229224303537901301,

    33502995698760097148813524749, 507733656730295659336969936147,

    7121781655369327442384836362815, 93053331015220741117954271462427,

    1138901434297155622985397308203296, 13121014689974854507957192066635222,

    142902196124243610444270856823486925, 1476900938482861335402649753431280097,

    14533687781813357153031163357787067162,

    136593468271698571243443389939628590154

                                                                        n
      Theorem Number , 231,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12, 119335427990, 279350578975291305, 38337478013201343576720,

    1053922068201547458460584658, 9870302960119106608320515237625,

    42030057683601061895373781418672625, 97241264497096510515285237495278713051,

    137719299059532885809027021989785908957530,

    129934370122289867245600063080132191721773253,

    86941769741833635384059870748154516140511925139,

    43282737769339805909368400842272920960979157846505,

    16646860134523776640953654192586061641328085807054225,

    5098120367654272799545662880661536154830750845112446955,

    1274332284224852531346380762048823509052304174471073137659,

    265376573796336437851955694579559605587379374641113854814951,

    46843249014570255746525469178723896393377030123673033454729060,

    7112378637987618928594520405846356451458027073684887745591557100,

    940692951859371988716013216778029489370968879995731618929084128540,

    109571711555717429473791397819219273850916588443305847164305617875393,

    11347990630686088797981970498716932022244772478095131995210253684902602,

    1053812172846522065677494936619406966193331006280068260534787668513681315,

    88402896407499712744364553138641877547649992972971292823580725936592098956,
    674388342794649570769448424030149721555644903345719417575161470363121584\
    9650

                                                                        n
      Theorem Number , 232,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

95, 320413348536, 595112857315074850, 72616801361071459849195,

    1852753551767206482779545584, 16457663601918945265121223430270,

    67327691288559322777470490196222289,

    150902336121649863495513399558640373778,

    208237634436076535148722460866005677344805,

    192234825559208386802362899342455420855700730,

    126257588562944234136748622686293869102423703030,

    61849360422967086059836821272446192584219122100297,

    23452875700493313400274206994281072042765678404815512,

    7092646079238989804015465255261169484470688805540319375,

    1753014069662105883836398278313218487635853874759922824175,

    361365476322120681030945035322752815656974485522065757447965,

    63199867301355675807854856575259105128176311828064636285058091,

    9515125877348205037626276575717614584250825914324944261117771741,

    1248752555723146650493677303914370675512145626169805871781206368920,

    144416075225559058008261056455752441135229508802351432120255443834049,

    14857731727181704728255618801245500444728801618394223303543881222485365,

    1371242945432879351525759413508798523227546931196477762036475018907842905,

    114370512251994549815635992889334579797241810210687815585744419745304905041

    , 86778797361898869021961665324963330640295272969705840904194743550690294\
    29125

                                                                        n
      Theorem Number , 233,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

77881, 13757069923965, 11251418755793351520, 887941814129938249347465,

    17082505020937134009417468670, 123928339206122990960109653670786,

    434382628965547451589014633362141205,

    860820202840530435045969941609070791655,

    1073649534058946739621351359012357537342955,

    910357634866138759166770081783163888640478864,

    555923968789848979790813084830831739650945062600,

    255626891811760522374927716258304636406144200427905,

    91681485754184347148735810962707382620156768501482309,

    26387192103903083640062480441138968781124588653354069490,

    6238630144319552849973640041287093838095264667979452081519,

    1235462414234957871653289962619065665137672852525407399512366,

    208331761973660552653325681983659032033780252054442329232350465,

    30336249995912934424423543670101076586770729813487179633865572850,

    3861006288007235999309410556073864234947516863287714651314103978040,

    434045481313989809842439884234187000369941452875858623281615144700575,

    43497309317898403089589796211620051223141426536026759426669936775006595,

    3917464943359019037615617799654662222129375644850357545310822883850405030,

    319365885834209046473015090717958196237421139113132824171774716879311000075

    , 23719202998273887338812597158224038694119557310241199737991777364969278\
    635865

                                                                        n
      Theorem Number , 234,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1292855, 81591292945479, 46659704239281092155, 3015952121096447365060920,

    50832156903843718088160454881, 334971010087398712642763914177440,

    1090334670511569939115126992516877525,

    2036269888702511195391607766070553562805,

    2418382817656326287150656315326118870149349,

    1967588191738547264587631555827760387976846278,

    1159655667797931747648318835380769807283272605510,

    517005852508079353680491975863328286344043692090064,

    180440891701804690627828977782880632062728612389875350,

    50688282036143802140548541788491173818369225367918121141,

    11725715089913301163305633795416777849684114909644205852061,

    2276763515670299453861058723863405337809070002279760913308820,

    377093506581919792719541894081534695317934040464337255727600481,

    54015558781450702608797590440591441869410681801875517613849492100,

    6771594268260853746764383542768603148539778193443353715476931950180,

    750683175809940729473596087596209956538175753433427817372345771625863,

    74259547515379343267941070971541704704314840983430445587783749162002535,

    6607703070785017548117167396282458176661103786887542727438155110059585980,

    532640076205474801173933869624253982479405653843862680229688738807769345355

    , 39142940180997255592778777016162394058994968391761472556622118199555355\
    180206

                                                                        n
      Theorem Number , 235,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4818156, 194405732782685, 93984319466654070375, 5520249249384641634777216,

    87263216115034951583289832410, 548709318280717724187317400113691,

    1722480016111424385323393501500897796,

    3124480226653958700715735188127555862850,

    3622464889775305604303414815682764612880001,

    2887810738861251480396602537335426935118327515,

    1672463389470964402321093427528513501834106776623,

    734324294789594908970747842454451730584021539849562,

    252855351005614883867719522478627620435115270821453455,

    70182169746685089154957391840435468017104963781961516091,

    16060883055664145086945328107346270758195639439106362163750,

    3088186469468513323259589940030873444388862030304976237605852,

    506953722368286000849081843684870017062718337303108807236366152,

    72027219994666348978247138108795016377419680442513167699404547600,

    8962078700047182028430655352661073804111733426874931684351394028815,

    986643959250030494772344514851761548030035430975713241803704641266850,

    96974677721820490571159171964725901458084878108617498661962197454583851,

    8577294524922156533898065239149404099333725804783029997659672333186115886,

    687539877209548023769834653579891384343951262809778965423388962894790337900

    , 50261371241592226927355126863847539090754376798013329813437086713445358\
    267001

                                                                        n
      Theorem Number , 236,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

17087631, 456945879607550, 187988686570965839811, 10059192960497394698513964,

    149333428450086958628520569455, 896691318437971484293770372818109,

    2716000142980888029430620480536990190,

    4786862049887739619041124997538629265817,

    5419043935341883412135958798454267940058062,

    4233743655423489941976575143754403631189980150,

    2409734231883394190195079970155168671502663983786,

    1042115631316253338957029076803038835131636850855590,

    354067407229695158455506808598329866482761644791419116,

    97108324559614581151413917474722779651971531254827364756,

    21985595877775690954164829167907742530257621866265217902960,

    4186506646710762560413376345511539291580522046328050424542131,

    681193866743348909782764204229444678402346573605598908767615220,

    96000889715258648585715191861374174851467695681391770756532221052,

    11856125529734288429546114939411699529696322518983528646011983934988,

    1296265440415132268661665025414597241410672874507274661374568302496790,

    126591924985490533583925637681072035239074639621549428962819892753229030,

    11130183295361785849069476614678759962841522687137915881987910786656983100,

    887204177593123200646267102278361178612177653218871138849498486311252604810

    , 64518658705026676881423671821121750413485918707682394971600159776670801\
    339324

                                                                        n
      Theorem Number , 237,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

57993750, 1060213898657190, 373476804056468063111, 18250768317670703150479990,

    254765324314286443269786792765, 1461922507282986031013325625123343,

    4274614053474516809715048046407960930,

    7322545493470913419582615059348528601605,

    8096288375862584802313004721199962909079875,

    6200202743034294747827351607863535161148523250,

    3468725512514074845719368582452810580701362935653,

    1477685974132505709688314028469427182278449758743525,

    495425711060137116812013996377351130618228012246842900,

    134275997521566248221595472635686134687680051625434880810,

    30077886698407863369685626389541227821142211393949598396550,

    5672363109546966645702403237879797369078636984952146704147163,

    914865823481479164574603570909199781233850572312497734213523245,

    127895614463199477478434399026559128789926274910880874576615024975,

    15678116164816285297585468711435851099279896381668607046252095080570,

    1702385329643171315891424094898404746111122749313588048784529880323668,

    165194632561243162156276102447418437641936468479229594103215990172544039,

    14438001562388764849896582772932303381608355481775021048677506020444407170,
    114448846915845385463037068538333466381547553733665383584459754211728434\

    1751, 8279553315618322228116875167654235108893499974575511872484303187962\
    2987373290

                                                                        n
      Theorem Number , 238,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

189221950, 2429716491218201, 737121676420736802459, 32972696103131776847612230,

    433315926762807670442519052475, 2377942582906828682577426327139935,

    6715303351730026638134569030858914640,

    11184549371166043129655089077466904761050,

    12080904291124360074533365464300882560823075,

    9070199722548455086815265160912366881402114580,

    4988411086482836921576782961414110318240754192840,

    2093578449891102576922621079110732313750442651556587,

    692710972856535968627958696780636891849785334584579158,

    185547178421873659531120322416509058241243645258130134915,

    41124265879038134595929841289780198883989146547403231548956,

    7681420028601942460956242355614212138541970763381583057491550,

    1228087873917390232585976284120373792142070688914482221154594111,

    170309468430794321171733904824437990606308404254329023265342026882,

    20723488137680802944776266810211230634251515416535102754998258378990,

    2234873645919279782635851188848379944356482003902706984071386645591255,

    215490824337822444691907190554981392224010904062679715440965087542974430,

    18722559017064928248853694237279020257923439988898529886623109214438784154,
    147591702657989169527724054517640452422176463588836379929536812962547733\

    9964, 1062183454649156489371821191092739088720366341748450472800108882542\
    12364451995

                                                                        n
      Theorem Number , 239,  Let , c[5, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[5, 14](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1816386909, 12323221710781653, 2817420858847473434440,

    106298293387966638165302425, 1242417156096987267721663003500,

    6248847548641010251587218936434719, 16483414899566357975585597055972108989,

    25977373755934234275342019408699722957690,

    26797728069250508926691682941800905146866846,

    19348346623798407333637960099093259352390093060,

    10288076386350768004445124867376290080852567489292,

    4192144825654267684224623490145715996718655286454095,

    1351295685543158179479986996233892975898835797595686085,

    353602608236312552062687267495337791974725504971908278625,

    76741637684346006407907239461775105660365299461494078262640,

    14063626081950558320137791379866851024421627544038860070380151,

    2209703356340294067629869323609424322784063167788701912761212947,

    301590112205294173073069798980369482949439433009745649760357585010,

    36162445886588292773560596549159775976572349693162800435818919497405,

    3847154669029508349833258764976776864287877300103544152051193731899680,

    366290652001231426373054788995990353381797392787592770368001057694949959,

    31451678601438539926555179365502393632176053716746833713086049453859961586,
    245218417868061844627076229671308537034559253522130958489490623381791210\

    5305, 1746625461639178074326499525056905225560947189705061224598121480334\
    50405470900

                                                                        n
      Theorem Number , 240,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 16](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2983, 8866806, 5482409986, 1426501794869, 208844930262267, 20091777722658976,

    1396706993838855593, 74747417111011421862, 3220819368464382814594,

    115510680802671653206549, 3536609195153397110650358,

    94303176905596592984625762, 2225369830055208024560884400,

    47088145734664957822212603206, 903208988862159452103187383333,

    15849579775677439588100000228791, 256444987009780670293133041592644,

    3851563351329239014172900521171660, 54010664748965457227121518437572613,

    710783124907922330226965305222145961, 8817813841648639976806375237945824575,

    103533321675760457860183541242309203138,

    1154611821521337902556406788691017858600,

    12269004416083100334540563940357497925981

                                                                        n
      Theorem Number , 241,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

112, 7919323078799, 90773079333959750255, 47137297889775954518319111,

    4170942046230905154763157954945, 112256551135763346922175214217517780,

    1261571569635066114328644880726413340048,

    7203661471800881810263629471900106821102129,

    23841652912335713250361125133112659795060307855,

    50222844989748714652757595422963410176588614337120,

    72172743588409511158497901169009622584189146541404859,

    74615598261956954198397944008172175984400778377154880093,

    57865603067568217598558790082077690642496620072859265083262,

    34812153486146992506400126502560491915647400834116031526166522,

    16699119252863163121312377325740371017072363782473690494742716350,

    6534626491838660612075551530529486270670966384159509452447370437933,

    2126446779269819270649028656544730819918535717787967053538946210810880,

    584918368247460109216698177079986311750303709066549576174581123706286367,

    137926236267074662310440336631647496875731190393007951887384302942286207207

    , 28222972855812516064723993727219135518535402409258032257339643131996088\

    354375, 50651144742693586096743998539672075996944738830310752352461096705\

    09835376755542, 804781874791540923215753689956868824073957202797928869157\

    233160174030979332412369, 11414996812905538245901877664085894493794142082\

    2230354356152542919329340632489369229, 1456110449302978136596229694065992\
    7499963575253806976071996861682545238329208092669064

                                                                        n
      Theorem Number , 242,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

785, 21046639714010, 192627057967038364893, 89108073894252773919473201,

    7324029179078177660698715112517, 187048190765242718480780343479512095,

    2020078167333466978998259740389567024955,

    11176292370278371509482371774870801511635560,

    36045581193536246256444722305820135787691576706,

    74301494003430447063450160333573660258092398688225,

    104813549135727687649664395563445142045432388707474480,

    106631490634108902893522227801707467801128070269409991427,

    81533291142978277631234497600955552355835164685616621987578,

    48438610082587605942440082460124465654442564971149780082034610,

    22975667118890065215612017526182594715743799700787646385394062771,

    8899876977475701436722495970561652249045271556617342197552569112385,

    2869515901835895357806351159627181119675855051538710900985629394348022,

    782680511062880727687580145658001731262859313054206184683723871746317455,

    183133872658576062824872421419341347324764317109816905165536479510787503599

    , 37206267889204567548123128982329792573278414960496904343412270574449315\

    671731, 66331747110634685694633160424696768135725154541946186227837110626\

    86929469889110, 104744178693771900948803452709792866179896963607054913728\

    7840834031625675997594341, 1477152628714136398418186475161003542992937808\

    61971722226056337933419522538649047115, 187413582560572881528691775773923\
    56408136874076614765944826777419660275218348603769000

                                                                        n
      Theorem Number , 243,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

25335, 141149937013431, 847584044332732319630, 314050556291841839909122640,

    22369385938516959646942475139915, 515676909405560393042901639829432700,

    5151029972669327372033177750229218427206,

    26782822738678393315692935301960714724628850,

    82087979576703051895384509186612579928942994556,

    162117340884680517435765781792216396757139201751925,

    220461300871599050248488332134167320648593368060111026,

    217255738096734376270999049856766143523526064694849748993,

    161531366022587553471871824173032492934303132212682973785410,

    93605975190627752471236147469620192849298002350182897023743729,

    43419882479467140996476519451878760988097586500702423536968669990,

    16483527596036060293250640184286150691881227720749804740377695020325,

    5218128405339930041080561766165328520705371597549175395613010344264691,

    1399622811853799329393089710807530444486487309484129449537155098040372500,

    322480685347799911724392948277646692295450757217170408108269575526951627755

    , 64591038984946149739415836463732162309127867037794631336658893766951950\

    618200, 11364457876903280383501788645087666816554273316091773507694601952\

    466642918917595, 17726619864777020409418595540439742593299897953300442133\

    98826731727123691871493573, 247140310743979400794688813888330648944775072\

    977945009956804779401812243131239663250, 31021107765508892097320301612041\
    200174912268295137125801116805637037190221035428652407

                                                                        n
      Theorem Number , 244,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

123377, 356850097160469, 1758318460513800289889, 585623639488212638042338000,

    38912522382324298115275294314082, 853274261611052583356704143615020563,

    8203186154737388977896980451992738460242,

    41369827075431714277222994776973450563881510,

    123651994375364355975215225651250915101778433270,

    239096116576704472135768524744011588171563024818716,

    319307293132219712856693194346409333073355239704259885,

    309746041009904434070349761877986677279612983853770046335,

    227126601162840690411839651564528479576337092758092331577454,

    130004317230649816696066630001568333366647699832807972519574440,

    59639937484922820946325737810060814335015622039078320206517884083,

    22415853264434822513265208859085352294987421426829797964085960549812,

    7031834642709209861646038236674810054124900895409649195331421679179440,

    1870466222293365100032714550831928874993997339338813621085221010502896697,

    427679427299895746152398485892159775454590176019170007781853739916506731625

    , 85058226998043313903199974511146601382371334328465672602922690877802860\

    511444, 14867739341101916892910520400324400914152655494548592770180369861\

    606270634412642, 23050016253489700762860735365802744869409352810377433788\

    85491976220178098889850281, 319531075887717021942017817599291086338424117\

    645492016424722462138814745887131405034, 39893999945702260111129833920373\
    778415466535703647448023761515476379619991016066160340

                                                                        n
      Theorem Number , 245,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9252757, 5287309927015520, 15056232958955322586356,

    3700722369312154082927656173, 201186153549819718068497431813836,

    3814008943846888265295189493450588414,

    32783818220686758340385890714222057455465,

    151159351345588672940954488291289337433404821,

    419617523404832697610218558665293903281785349422,

    762347801686530307997872530229493665274499801004005,

    965045983699904758799043284242293167022968706291280037,

    893516885336171922396154670999006144285105862806571582850,

    628816845054808626453154651092933088319386288899561640817864,

    347001702344214099631131391890953898776796514861983680498015071,

    154046269174582833404362665677922963170152318006421852322150203310,

    56204584737723786408777800617660286392078222489309052628139980694103,

    17161084765629000558972219381390882307156477698421081689277656666458875,

    4453264763980408955725071981365865571798448374871020083767699183224369496,

    995312051720309461147532146719849353371780237046804081292575489538563488648

    , 19382945212550014064349112369676003719646362866928823493710504857405674\

    6585621, 3322541364557097388755548385247361607482491530190251586081016202\

    1774530248193544, 5058271055768557993724667776673617122572458922606619709\

    518054149322075620542826440, 68939551054762816986020592425038590278496526\

    2460293529895070062906873920094418335050, 8471332500224700711384971011803\
    6870277143642817101804115122885927193901326645506906210

                                                                        n
      Theorem Number , 246,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

34853507, 12644757688676861, 30397547579650435054355,

    6785412883145296919312486902, 345864866679616354211440745510990,

    6255231304180273744095420066962643675,

    51846370805770629988575852882856826110923,

    232164577918265983745741159890356833936651345,

    629094240486736917711960104420498253279395021442,

    1119804532010700156695767558804045240588024940553147,

    1392857931702036141246713774310167237125847865506968206,

    1270009416550138127603659838923592372425085263360255657023,

    881772348939430193994315127837946648104771376911315300877375,

    480763621523093651041015391080375458948855960610825889992363931,

    211129781330321752474340067277428022548423124894405459321022779767,

    76280668672688895111972886926055901547854354130027842520721213699260,

    23084011839351665860919487833823277903378415235817630204428828368712319,

    5941481233380037066349195430818669747609010571634686990653583605990423105,
    131797604722762505336021698906441703071099744199709504970876382759506168\

    6705, 2548864217958153716636637043962316906531740770790242710417068167633\

    02551249496, 434103158540268195617047661493956766582391426337619260453577\

    11187002847282070160, 656919716796784743011100395772987355996840931714420\

    1543180101517861622701359273293, 8903018260398485142979423547836742238453\

    08512763505950681753281147121516476376885920, 108825873300351931756423643\
    953778242603295427862733899412780056661781817642594560127577

                                                                        n
      Theorem Number , 247,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

125432887, 29869885635827034, 60984649734418615949466,

    12391617298596041532161211370, 592904868478897894962381590566622,

    10237052077494748557097829353375772553,

    81854273538235116584712628038210914967353,

    356086574039730680613460282941435803377046450,

    942050661681229564354548239544902281942812578485,

    1643241716414437880796210819059508981775684714928525,

    2008597964688283581154721852311147621800800788141931425,

    1803781750267943295408035959807809234971157602698585520385,

    1235658704250729581093670704435806188570573505133166941686947,

    665689541749696080777099936927066647506819777266921137197675000,

    289210037813111755563996423533427668192535348913516212207777283381,

    103477109560821328883223160200560917826361792807507697939363951118976,

    31037252892397104312842398272799275482409611695830464120067348291035089,

    7923777051907475952240525617273528046813633177911018281092199987468305410,
    174458000039752446075689517503092795692999181044427747999309365432134742\

    2215, 3350586711056029240416526413916186741125846922582206490388151348500\

    26109297947, 566987408950521988246029450343255328208922623867471769057656\

    25339406066097168163, 852883885592826164029710479524556049687998139022542\

    7437068755235463431233297332273, 1149428514159357718210962265935545472945\

    162135587744070589815203198927393821961355837, 13976424690170820918527997\
    1569428216758356527789294675543923464819489740683732513003490

                                                                        n
      Theorem Number , 248,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

433331947, 69732548961754298, 121601147574518193545614,

    22541186201385993801856584400, 1013569302981378687940236979255700,

    16718165797232434811396549621780136798,

    129013996449708193208146720334612691477863,

    545405936193387676732177164109137542682404905,

    1409073547330218096550009198662222593813893006084,

    2408985670426967097518344759719377048953650652864880,

    2894071465875505644140407811346530667438811008085283311,

    2559976863844723788498256457135498020264340901677621600246,

    1730422223599923037612083128474012511715145719145918124526818,

    921199013616828076132278029931405721953379899983121893343604421,

    395953329001017761817853792877349059620877143046273032216372919500,

    140301455209567840833423050134149400397462691704790904218232251394394,

    41712055282226984592311336247055872084442201928862113017944554174382125,

    10563111023539457077173535590617298782094191412691274555102603949621425734,
    230839463251083770656904197423124081053621888591964468033567200391314349\

    2157, 4402940814303639200972786442475156755117265491216239179077275134914\

    68414307190, 740307809623639439019453281896073769058912027020084087552827\

    44947987289516106752, 110696885978765414378282657079356633316191785588900\

    86219270491321383239388707696375, 148355270272105157044042953460697426845\

    9265971000475792137709302809025513149957225489, 1794501766007130036646438\
    48084791227071165296658527645145613396913779942204200019207593

                                                                        n
      Theorem Number , 249,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4644733784, 367554472062856735, 474965369090658987652043,

    73739489514564196224748577310, 2937892710285081411051360791438115,

    44310887722715584318973997121611004350,

    318916969956838248110662448642225494477925,

    1274340360915844713099380882043934800141166138,

    3141754840862022918571950788297088784696553208155,

    5162185608551551618309891652532334179747383708393875,

    5992972492463048373752739767024412528034359955304892143,

    5144860152150083529637075136183441478813615389644511169915,

    3386885902544868007670700804156674375139552313797334289257735,

    1760951692098250629816406275942891302027820198116980664615899498,

    740986709864111188373344525290427154948836268069192663915505363009,

    257552832410187335261273507902578872024876543419704766385336480708795,

    75238606277688516338862580209720250524007697450855680550385738177719890,

    18749096624237974256081680376043647428616670227735379278812947996470129254,
    403699728047715051887586084553010289234564475926807914078602118801972249\

    4490, 7595085337116589089483256099481983842922156464316044545406055649632\

    43712229826, 126086212713128050226976613269998705950051864418504554763904\

    084935336039792953773, 18630817396273768094780368719191247681656789601098\

    415165328619700325459495001417915, 24693099466163850192269021681157607447\

    83597995229763103752040015886863885275512721985, 295591960175611618932626\
    151207717251419237670927660834487430604810393104381007229481763

                                                                        n
      Theorem Number , 250,  Let , c[5, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[5, 16](19 n + 17), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

44005064763, 1858313721936769294, 1813632706844035355511923,

    237686576016357750553470364035, 8425603316176375664765442747152979,

    116493577732726038094376461737780879293,

    783256746774013144131592477917578779580103,

    2961680049493684083162610790846564265223891371,

    6973736156273137946255787516127384023971551962395,

    11019617555725601233262493608143849276020300305858276,

    12368770409854752282609216835547300817335039719373206872,

    10309465560937572142893455632752201100160112219596064373929,

    6611736611187986040491757617723228814790720429870879833296578,

    3358341602504089331104609835345289119415184239763283808886693985,

    1383752171245408460429740783244622038869310337033195896113530084226,

    471883837134847329867203242097244320872899120558235047260743957161188,

    135473938855646235143336075514942325804065080762709046367397890843273959,

    33225102156066032238392800534317331327758526901258082396868774828737481548,
    704949562641825702561561149479344444313089700381389085837710719630640983\

    1505, 1308340609808207976106754279418926293832010760972358086526170179287\

    785570244951, 21446820155549677216672822386719418480307831887667828496623\

    6364424034318326982980, 3131882919921888103131631717648337757226292168937\

    9675213387436141531312647488944649, 4105428223645998696153462758434763146\

    590383669146848965865001666600045674361596949465, 48638629691081411834304\
    7748678333742938794350413324334143178107427248162630287813866000

                                                                        n
      Theorem Number , 251,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[5, 17](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

90, 36630, 4737915, 324893016, 14584338000, 479844695520, 12402843931800,

    263823898666860, 4773056213425788, 75264046707398400, 1053980151625898910,

    13302947869002166608, 153144011349850243410, 1623767159413586023218,

    15986154718343961554175, 147140843859685815419880,

    1273580711492723453186658, 10418546345667855135678000,

    80904418307072439827353173, 598663549359072260393213340,

    4235486575105815055051581180, 28736438716293622304761575360,

    187469612494069111076739344670, 1178790493624852323161033109360

                                                                        n
      Theorem Number , 252,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 17](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

54, 612524, 721245546, 287906817312, 58249953408606, 7284493331873676,

    632388090975539346, 41086243863913657264, 2104600580497855628494,

    88284506315811489251928, 3121040410783484782582719,

    95085128687372032317804304, 2541291945063489033030948708,

    60451739099005463235604747848, 1295290558060723351056531611725,

    25251082391086442727218453914056, 451685750403134521874043416408626,

    7467891004534706258992310363558708, 114841619379576045349435895332357863,

    1651676654197461142054166885911297736,

    22323888168187259304056379427581702512,

    284764501531343755928200578177169434384,

    3441283174530840576874575969698794837446,

    39531803491948260970880833473511054355424

                                                                        n
      Theorem Number , 253,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 17](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

484, 2842749, 2559878242, 872073717676, 158294339200116, 18262635305028007,

    1488535816754694296, 91886406809910146046, 4510311776126377165752,

    182459900684230694614568, 6251127807400713081406800,

    185281227957116631540603868, 4832728540144115978008248088,

    112481767496441466646043606802, 2363234380913563130571868864568,

    45255538869697479891920026241000, 796436551272237856946072102309040,

    12972214884672360849525011481798664, 196752214263815752808113236849612456,

    2793765617851877625566138672999153664,

    37313663687375334810432856708483895200,

    470719079700040029270927470148264174010,

    5629657224559627679952240780449449905702,

    64042716563865837479484062305313995577932

                                                                        n
      Theorem Number , 254,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[5, 17](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3521, 12318504, 8750602800, 2577011097556, 422525038520117, 45158428024439040,

    3465248806453194386, 203630137615310193200, 9591692831006376932505,

    374603338441101063211208, 12448197148650655069130954,

    359198129615443356235928004, 9148600504267815875628836666,

    208439200415308919906404360264, 4295740175103594243102885804618,

    80834669316439127767181338348564, 1399982540547124025735953237258168,

    22469482500507306430770766418584848, 336198237906351034546227929976511356,

    4714025808407257917657688693393523124,

    62226477612778787062097749049541251709,

    776446152194717175380439705835375373680,

    9191251752153557184396767900926363541946,

    103556133576190649470559132177645416175592

                                                                        n
      Theorem Number , 255,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[5, 17](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2515, 139239864, 622909798340, 792055691765386, 451705779268242390,

    145450098296650138000, 30304712337723074199212, 4465091291000509343862000,

    494930782367731484217767693, 43179813642146759454003391320,

    3068385839359710173644918431870, 182400962316038554934173595551312,

    9265867774765972035375008201284760, 409275573830904252734612746513859232,

    15945634405848042360040271019701216080,

    554604394424797982414599789882410981200,

    17396765202040725552484967368685749134234,

    496470692568861443665366352575944297384820,

    12988022616222151604778762922252106183217120,

    313531709718151921405764842275450359653493840,

    7024667206105278125574150928092538059691064555,

    146825516825601010063485706316590378003401896624,

    2875998100273226881631163904712378583464648090820,

    53009952758662170259806816591236211138948933660912

                                                                        n
      Theorem Number , 256,  Let , c[5, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 5
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[5, 17](23 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

156792510, 1735167113348108688, 40278527816764485117522580,

    55513336357055212371270224733008, 13527495133212779796108206315117663415,

    985785375307269829700240934998986058270920,

    29063291513867081876108566636240578531418600072,

    420510548670434502138889983615341182488874702292160,

    3408836768542361328778789512358795161222856002874417585,

    17035620605509302595568887771900655683319761537936948238680,

    56388240050126165868031521989365921951276491958852017415665240,

    130675807501640752202128110676228311561098605585377613804010899658,

    221552906010850737049790040711508564782452344525463728448241687256215,

    284762161605787081524458594462869366673334022523584890782470169559924672,

    285703706593496801239962480575433895052261931053355149692397393361520910320

    , 22928815371725519261841980739584723080638469572156579451426073407586392\

    9709280, 1502583491955354737732833698853504833022966248293662305733350217\

    71453864372967449, 818349701952508819458711070634065066003818593344260245\

    42658700477298595019217812160, 376080349284370463027896933675933396546218\

    05413228179261658578430017906863582023070884, 147775037431663831175313530\

    23105756279557688120266005457907627443770064435816348764462400, 502257784\
    412387402757367004630258902779011297324803235936630398189654459062836459\

    9754622340, 1491732083633515715910101733026648003065715922546147045377085\

    370397532908078017133187373891240, 39068586961697650683565185000290017103\

    8839071219460378865217291190953603435224734663502064611500, 9095943542294\
    458293080340738255726488414563501586071953008640931052433820952988411617\
    5095539456112

                                                                        n
       Theorem Number , 257,  Let , c[6, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[6, 1](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

14, 6231, 498940, 18749232, 445877466, 7729757608, 105871163220, 1205640225898,

    11819759827135, 102287169408276, 796130703755124, 5654162780790750,

    37062191269608172, 226294879789289988, 1296837216423749490,

    7019425329171859077, 36077258983501036200, 176869306008162651540,

    830340308633602114514, 3745581971564446921620, 16282945041427199960266,

    68396639197541495597320, 278248952378837139092960,

    1098572755072891168430540

                                                                        n
       Theorem Number , 258,  Let , c[6, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[6, 3](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

967, 446112, 54704256, 3315578912, 127085416992, 3503479485792, 75058684074720,

    1315047445668351, 19526866313143584, 252260665733372384,

    2892229075074074112, 29890327267633348320, 281926371120772619008,

    2451535330662755856000, 19818464428632193315248, 149995055491580121697792,

    1069161451818743226390912, 7214139334837798115991648,

    46282544012222883911293696, 283408445815718244391109376,

    1662031887772209759834423168, 9362565435007303004512384831,

    50796474839123094740908004832, 266064787327736376561354534432

                                                                        n
       Theorem Number , 259,  Let , c[6, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[6, 4](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 2286, 299776, 16463818, 540897280, 12507329650, 222922388934, 3242426068474,

    40005146766688, 430443765118080, 4123222014994070, 35725315479183146,

    283518377676436178, 2081819408868240460, 14261452999290965478,

    91778868957571045376, 558096415665587739136, 3222702699386186526830,

    17747324835952744320650, 93553515132669276709670, 473601879687938792718930,

    2309067535124617182125056, 10870076110416103684879104,

    49520449020011176883281792

                                                                        n
       Theorem Number , 260,  Let , c[6, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[6, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

54, 18582, 1608078, 69979264, 1969607002, 40691764566, 664901736350,

    9019603799934, 105053846646784, 1076657138930970, 9890759560669234,

    82628150736912576, 634977088566876890, 4530732671351480832,

    30248327840050793534, 190175492751601751922, 1132135084750468566646,

    6411532146790225716352, 34681580291559816029618, 179819674085890669465600,

    896432619522160507010394, 4308486624418693928355930,

    20012953530441296135725056, 90036280064710115157534484

                                                                        n
       Theorem Number , 261,  Let , c[6, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[6, 5](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

495, 20306560, 43696145536, 25350438234304, 6637900902406208,

    1001933373890370496, 100234950342402247360, 7262616914342822429824,

    404929253294509769892288, 18147553420996212212267200,

    675441375796262689701082304, 21410766002101045936240465221,

    589674380276431399505702248832, 14339122448698005946352372419840,

    311967996424204067327599614144448, 6139960367878309448951499151572608,

    110340120947174443828317982427952512, 1825009883718066782354507792597203776,

    27972728352473932144961597718779157312,

    399693048446113759256000078300219366208,

    5351842149711959254437699279770125909184,

    67462069202966055360563385489550098811328,

    803831310370418396960079800049624953948489,

    9086508873840995310663139730588686314217280

                                                                        n
       Theorem Number , 262,  Let , c[6, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[6, 6](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

64, 339672, 180915240, 33964613568, 3366545794560, 213865631951040,

    9732897207596008, 340694753692877400, 9631342749620064960,

    227768009852346954752, 4626519571407336993960, 82384600247601036147672,

    1307082662974097776282752, 18720476570226569626033920,

    244664192950066531977215592, 2944234963492566922251315072,

    32872036907035725798992129280, 342732408656642356178773339224,

    3355770803151828396137378688000, 31006665897717295334906391538200,

    271518049226446465492766891675520, 2261829485226379499487038244651840,

    17984203492917385962122732409586920, 136895516121974600888975076254805912

                                                                        n
       Theorem Number , 263,  Let , c[6, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[6, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1128, 2367424, 876619584, 133350855000, 11467427466240, 656261506051032,

    27541330443346176, 903112168268072832, 24185338913688238824,

    546322618406087364120, 10667505186090703858880, 183518058598982451185304,

    2824271037437836606824360, 39365870406026639608373184,

    502065544305973889781535104, 5909476829826132303631931520,

    64660739903356239779929154920, 661817629851899933657300654592,

    6370582198597945022242834970688, 57942905454001316733818706573312,

    500023633995955054174288629738576, 4108949592509760455773983749388480,

    32257192808623620039081354840159360, 242624137314713544822455566989241536

                                                                        n
       Theorem Number , 264,  Let , c[6, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[6, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4032, 5942760, 1878372312, 259660355712, 20901636863256, 1138692562349736,

    45979337156976576, 1461255455945197440, 38125243279496010240,

    842329802064695445504, 16135289668009322576064, 272965790146210878268224,

    4138914131309273919994712, 56929366988544445114592040,

    717438868281775554323365464, 8353410842774182980529161344,

    90501827591693270280004787520, 917933945885046309749683963344,

    8762271314765737328257411445760, 79080975037844475086552149056960,

    677538827135591993345082187534488, 5530408718184948409878847163813800,

    43144336534969726709444716255378392, 322604639455806240660284434308038016

                                                                        n
       Theorem Number , 265,  Let , c[6, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[6, 7](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3817, 279510207, 1129767219106, 1173943853506319, 526366223755949510,

    131065146053022326343, 20972805614859587432765, 2368551829279562092337819,

    201350936557532845745622184, 13499395090297985725653298779,

    739231220108499196258031959192, 33973111795891692416386520267945,

    1338842018762507258485577689695741, 46039547747966859732039112499177936,

    1401428874700794753529424257675092046,

    38216619077260985416019813737227696336,

    943133669023518170697669036559607978193,

    21246763917008884436048509949145640979339,

    440203386087861300960557960586278562790805,

    8442588330520534837151569953611846641759913,

    150741033411015946162145399278333617072567981,

    2518268527009603617062848570801303457749816377,

    39538779459907213153600146577541215908462582472,

    585764571963663091534629779807456924051778291277

                                                                        n
       Theorem Number , 266,  Let , c[6, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[6, 8](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

98, 1071835, 1004551734, 306177506301, 46775319777974, 4412006813238658,

    289710334979639658, 14300919256042904960, 559439740028396498422,

    18017589826076102464391, 491629397138483552956564,

    11620155278881956158313630, 242138408040403379005567434,

    4512130965656986110892840886, 76078326549600484012562408586,

    1172078371230751630007172185287, 16636686238211729538686702930614,

    219112280976301290865982079478069, 2694075213431399549773056279082488,

    31088835173910829816951969100545379, 338278702595404756591907714820376028,

    3485027858512792750666361311531991684,

    34118101654974989699162349323636180886,

    318438810563266466444483565982434778667

                                                                        n
       Theorem Number , 267,  Let , c[6, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[6, 9](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 17121, 7692831, 1208284494, 101244844140, 5502981576177, 216633621590658,

    6622416957140913, 164847588328228230, 3457285191672879402,

    62668951946654098202, 1001344636699923621822, 14324986279472054535624,

    185801800501933442631919, 2207714210042290498849551,

    24239273216170472825484387, 247710523053238828890988217,

    2370933575097581475369417942, 21368487220955624196934528161,

    182193533999721747022505516905, 1475608906878997881276254695938,

    11393409958307953884918731116887, 84133547976154153314103895969471,

    595877672676213511307078652982965

                                                                        n
       Theorem Number , 268,  Let , c[6, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 6
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[6, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

166, 234402, 63890601, 7555028181, 523362146109, 24744898464618,

    874389958316933, 24502940954313420, 567525031192594653,

    11197897997193537185, 192586797443377663614, 2939132597484352549125,

    40375000182635377185030, 505068302948232476755395,

    5809017752563315322410611, 61925166168009010645644242,

    616042180456926750142830498, 5752756741986700431994950369,

    50683241649343833878457372004, 423150491075113080173153817621,

    3360884155939335557911905634365, 25481879181122193482082436527517,

    184992776192962369909553925476181, 1289464610717614290782972489395716

                                                                        n
      Theorem Number , 269,  Let , c[6, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 10](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

16886, 82922496, 70778985639, 23375128721408, 4134188689653770,

    464999947202402416, 36920935637893362829, 2217664170461991729152,

    105799405967850473903810, 4155476304183158233802752,

    138098936193744491633309242, 3967357367420783778137350144,

    100233375368786717048999819462, 2258451601709817366626348872704,

    45913790032461297657363145474416, 850451282332885103137491749541888,

    14472167762887134354984675756771474, 227871496909409541196615537087936512,

    3340407736681460967711408177193003811,

    45835605239533201039857506350287285024,

    591506113660846835170091479147948699820,

    7209256468339310436294773957224534941696,

    83294981971006134105041263297709452550574,

    915368595703356927751498088521635250440192

                                                                        n
      Theorem Number , 270,  Let , c[6, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 13](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

228, 10863421, 32733349606, 27348422330285, 10319336783113563,

    2229095451512867330, 316231553616020410387, 32182008784836110837785,

    2496716403380450635013238, 154318704687204530709029069,

    7855532078011831963080986736, 337926750203488791641837988437,

    12538519193323311522438624045494, 407991477639068411989012475001591,

    11802554122235348992367632585349722, 307034153883045001436344163970311592,

    7252475917688454890500226981925398952,

    156844984119865535175859008534582234719,

    3127842718555955496377555841918442730761,

    57877930335471681642506166655952243400772,

    999187108356888498482837393864051247778742,

    16171153748367938678156547766726282157222582,

    246408904328330752293948747518394254772188261,

    3548584942460003270975090861331089229917156796

                                                                        n
      Theorem Number , 271,  Let , c[6, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8113, 128344551, 247802609012, 159468535979382, 50374351424806103,

    9541469915237775330, 1221803825349591571594, 114451866013970852975731,

    8288317812088632930071177, 483197563934139347900700457,

    23386576404207805047742548870, 962587699530305427043711518699,

    34347854966991562879753421525241, 1079322911330884097515098535273794,

    30257347220395804643893408161591979, 765010043469824596011273784661787057,

    17606619458897187159019927564586628209,

    371794079514126372233089999030080297561,

    7253189560625833632285661168267640364145,

    131509866609520326884967314835862278282464,

    2227809551485097341190701234608810862878012,

    35425173884837267854451172406513691309981317,

    530958960923309820016899687316891932800098423,

    7528975879057571092408873477337012742761866709

                                                                        n
      Theorem Number , 272,  Let , c[6, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

40086, 414900929, 659483488993, 376784967607554, 109565050930384126,

    19504504917148789857, 2378627689952585428824, 214136880909643101487938,

    15000980223285279303018743, 850146359470458904743503406,

    40151289905030946357634898048, 1617497204351975435373942249372,

    56627852724900139276210533473942, 1749358023477037970367008455759736,

    48292634851197561320006768714739633, 1204074899406894765313573798867301138,

    27360429477113202244645632162963657042,

    571034288751391892779749050313900962474,

    11020337862916548668500354435124085109782,

    197822259215571624464504784785952370529997,

    3320091374665711571338192906176733380910049,

    52337043416109800273547297931025672857553306,

    778080338835347132024072506151837748880745569,

    10949189186721017643509512798014842723034009421

                                                                        n
      Theorem Number , 273,  Let , c[6, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[6, 13](19 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1084007, 323574065645478, 450138251444586368482, 56854259621601669208589657,

    1668362910445351504511129060835, 17792023366230680486318283210693549,

    89040129042638926214372463165027917574,

    246048494024297088602761666870789992769878,

    419609912477717195236871537281799085536548440,

    478475741269633585143334281834357478303516098474,

    387371968012277920742432715213737709647225723647286,

    233228744549071587831906283514197371180694237026415110,

    108332745471787315870440873999096516058295526084794274164,

    39988867663997885989628422071079538120876916762108169481315,

    12020000762709121857997589264515762940252008667661183923324154,

    3002495887005484826076027612070221973467672349138290833028622842,

    634053775038311355995801732498124640876571799869821335809192392774,

    114867705110992890482142003008516892945930904491452172536347779288977,

    18079292540449542581990165618891738292135542942299042344491732516905678,

    2499439481568419263004594865287402548346041933830902091875465467726532219,

    306447049038559970016876535531162043838222000064243324843263896228286357880

    , 33604702833439300454375368494277238612149568106931954096819615671531143\

    399239, 33207883537338826322248107686704145548845020736247534647814170202\

    66669312134809, 297708427925946364737418678256111110922128688230880183199\
    206309213573887577271691

                                                                        n
      Theorem Number , 274,  Let , c[6, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[6, 14](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 72768, 81868608, 28432103172, 4858379513632, 507875612276736,

    36754217722317824, 1991783600889363468, 85286039782254874376,

    2999290526210750425920, 89179307218271760881408, 2292763772867007272836653,

    51883583741867365027132096, 1048400996350644759384688544,

    19142429950482469437330589248, 318959666478708846454053501298,

    4890765370260453121085483573892, 69506578578012831484980367282432,

    921219315805062216064478520688128, 11447758289377564277883076438152255,

    134011799834449443235404802663317408, 1484006565258203153264134758474528032,

    15602772801537811065480646254537972224,

    156268823991735722582778418266621649062

                                                                        n
      Theorem Number , 275,  Let , c[6, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[6, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

368, 1466441, 944971328, 238351614368, 32825017374336, 2927658627367854,

    187269252481814020, 9185786385589744768, 362085769079251296000,

    11870396708893784634585, 332207350540211027005984,

    8100279707924390297037024, 174912157660150339947686912,

    3389543797448978107061348358, 59599632322873907517117430008,

    959708633693482769264097799552, 14263851715766836212232364952448,

    196997411065245534378293320212154, 2542981371762600463319065541752192,

    30838671274312970068688921865202336, 352908236479821918865930282903465984,

    3826147539879520126955570039440194760,

    39439051929684770297558435360329677964,

    387727009673667859486986528396277692544

                                                                        n
      Theorem Number , 276,  Let , c[6, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 15](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

301, 23539470, 105659332557, 124809238748601, 64319732429109726,

    18502253949380339343, 3427636027766899004407, 448361130704037501675315,

    44126938975814274938716749, 3421593894823477129531820071,

    216409440614190145022668031127, 11469615192902569516050899051265,

    520416897598516286271561204068727, 20569848174952292144792694048815340,

    718475867360853261459555651424955934,

    22444040965256593434714360057351084052,

    633440910415524983644801727397901494072,

    16292960840758196483641399340788525969554,

    384805845182110082782427259916058303158716,

    8399811423689204569334424666391493890296087,

    170441082391520854713787425172574470140823155,

    3231140656845346282955017194422002778957008563,

    57486934579312145242035204880373380554396884145,

    963741798861354583892231533662116931124840414200

                                                                        n
      Theorem Number , 277,  Let , c[6, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[6, 15](19 n + 17), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

38927122521, 1421391470206939307, 1218915251707670396367564,

    142038599405564423045089345215, 4517212061701104201798448680152834,

    56429439758102967467768467385996509530,

    344779785712887977092231016368732262746755,

    1190420241995653893720269492681790497510476839,

    2570018599079572904934805141543535577646334915949,

    3736741339038480365541692080012140712078098755509602,

    3871401782104116466861365802171331193535319219486502150,

    2986762294939852376614921846886543806281054113686628865375,

    1777401116740204100385189107197008449443551528955640993450097,

    839611521953326817091924516954392881998205725013399885391778174,

    322392833129552384692745792999823092943029817874606022536889697762,

    102647914842755775300287881271545624669940279491759187356711134699792,

    27561825859285362718899075304725116569994084017582243255895378898589197,

    6332106508930626600422312705200587481107895684634594395746058581778671972,
    126041117829661986089977187088615015698658142078695825533792114411749181\

    4117, 2197586572242373270851205593605248625251601714421848465021715305321\

    74715967124, 338858670632340208077385804603661114774250200087707695146629\

    59458711127143337411, 466032642215242777756923855211018174127581603689703\

    4040994774616186454560510009460, 5759935534031830034260080834149810963702\

    06024710943558666603073436472781807186017772, 644100626549206867336469494\
    88450227682344151748770552073135340521939043023624491533854

                                                                        n
      Theorem Number , 278,  Let , c[6, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[6, 16](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9768, 37763654355, 2735277950407768, 31927969790672077350,

    121038374538822423192856, 209038987767780924824003601,

    199912950184748765770734481082, 119968269165142890106809695467955,

    49230639948912049558739020416638136, 14701120452244692222299942835964590715,

    3347400727767818318839901143155279106608,

    602601944559926362078724769463410222921052,

    88267190173044130711433715301779062671217096,

    10767651251166614896737933757394761171721637521,

    1115113834384632988310082060286131914657705076008,

    99620271416886201634606010772072849746715058926949,

    7781940292591217302502667957053510995226072859693656,

    537734317519283965826989341959622782845849797411087589,

    33199116867720816461546583069419625764840947751241996576,

    1847312777272156353763497877460279314546732608070938732931,

    93351279155843956604802809375371869970766606624053006911272,

    4313087807796535226457104335272730175215463728371716521102766,

    183290275959767455138885753002367422974118196084773226194547704,

    7202620675542473116862738666333573994420554344496329755913471126

                                                                        n
      Theorem Number , 279,  Let , c[6, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[6, 17](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

105969, 2580748511, 8423447838899, 9101451917094702, 4726007882433416915,

    1438763248598587279535, 289974962207486129235378,

    41951259929448886842734841, 4612978261187545646666799236,

    402179504388269812026717082010, 28713486149061358239939847374357,

    1721839549937715549692521956108188, 88507651617388455230007788663734292,

    3965317140313954296144942138506542387,

    156994663511863935037969934633494823449,

    5557128470805452515416725148724054181068,

    177611532980503689664717636583476392937849,

    5169403065450749529058310488944451258004093,

    138026057448714828469771613122513178157279538,

    3402771622504496779394899679163250083319265928,

    77896836677658511107459856730422747980423083575,

    1664198107871513126566294572792430884695047947998,

    33329832947596828570893768159510294323290082515037,

    628269183651798568674500873185833237906453963252925

                                                                        n
      Theorem Number , 280,  Let , c[6, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 6
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[6, 17](23 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

57595289, 1438350225088431235, 50185983680877533532335737,

    92484783940655870451891135148423, 28451972949741874790646950815309764477,

    2528265774475278321752174802921759019408885,

    88779026312926470398525568802193599682166258900,

    1503890745922393820600324844266517265630518128813302,

    14086594160595226446106638254416550463311360383558324269,

    80495735504394762485500786602436230161730266001146569582384,

    302064358153453800534739311140530738064828165922604671409336560,

    787936185002044590352628972164344701813791246404935857877346090225,

    1494544413736069921843228838298822325590436172785632770015681924399278,

    2137763383484840152459253295809822724350158191964759910432363131698519641,
    237594389864203372922640440786815052935205842460164884074753528667372328\

    3300, 2103653279543925387924215103026699974267141337095254279356567952717\

    788163181454, 15153884563918697969657978500168892123862732238546421629356\

    35433883320567459219459, 904268364310208819771537659852348819570805874709\

    998590199915129747718768597693018775, 45396998259438326711260195117016416\

    4304130244317892238163544092221868853649932489896286, 1943415729965461909\

    57538656989496330442436178924284551550186767658069514062131242416819617,
    717857974859280970737919894974854521124622725795554847815045576624753163\

    13514033878606439681, 231187813210173117750943399961944013312504196990767\

    95198636489402075455860292660741310400850956, 655170397035765198011166074\

    8128576966069433377801612391867747761210135930843278896994291596706810, 1\
    647339610064472660909972263640017684720345902629540775158423144465039073\
    865080268430010885421037669

                                                                        n
       Theorem Number , 281,  Let , c[7, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[7, 1](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

176, 55823, 4493984, 181940000, 4765653040, 91701877248, 1397056746048,

    17690376203424, 192566562425352, 1846678161396768, 15892890680621344,

    124525705154261472, 898516626301542496, 6026054874746517600,

    37853164055780466544, 224136045304973329451, 1257804824663159018112,

    6720761009996368157088, 34329234470862232368880, 168214041720267006076128,

    793119848533827445322320, 3607982494443694933797504,

    15873717557380197274430088, 67687704610436857982479200

                                                                        n
       Theorem Number , 282,  Let , c[7, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



              Then , 1/11 c[7, 1](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1712, 2859808, 781508231, 87233774240, 5542267406832, 236606890035184,

    7481502750726720, 186604462533573584, 3834762193020511200,

    67014214866180309504, 1019885497953208834896, 13769478559612629337360,

    167356230545419127533248, 1853069959618727322469432,

    18876621745191842954451120, 178359483391432239287218016,

    1574041130965356844096368432, 13051355958875397940356550320,

    102196016852611958885809219920, 759069990905393426065563414656,

    5368977061880668285848276366864, 36287450525294353703543566628992,

    235072686267723605046823093510368, 1463564611927166788221040112647040

                                                                        n
       Theorem Number , 283,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[7, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

14, 10296, 1297534, 73324050, 2534333048, 62189568000, 1180340044282,

    18302003085520, 240710336109194, 2759091947920000, 28128539080445112,

    259103148605044402, 2183544061795712000, 17005889449316840408,

    123421979381995459130, 840529561956555424038, 5402890357400059055938,

    32944782679064698746348, 191385981736728047845560,

    1063231894472949677952000, 5667217805013087635716824,

    29066793711752309196875618, 143823279864079461034368000,

    688121712235168160865625318

                                                                        n
       Theorem Number , 284,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[7, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

280, 81158, 7010218, 317542554, 9444250526, 207431251238, 3611599488000,

    52232056513826, 648368655419002, 7076285836412408, 69156604495718446,

    613930574237924920, 5007603621798581904, 37880308755797760000,

    267806305145162599466, 1780998342619137524568, 11202857011104277036414,

    66967510625033883648000, 381981000215802809154808,

    2086449056663114991312000, 10947650862610267201771102,

    55332835544957011274176874, 270061581740672512207917080,

    1275606402851052828039621826

                                                                        n
       Theorem Number , 285,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 4](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10, 57970, 24469416, 3516807552, 266820237226, 13072859346800, 463120468156430,

    12736871190207782, 285377762035924992, 5392134105701291606,

    88158556701425327950, 1272141673299383946120, 16457568539706349520000,

    193299037831567104412032, 2082658506986458849891158,

    20761995508394506569196870, 192901129814766080148512200,

    1680746968829792648572930800, 13806549750446701171109299982,

    107421684830683204194282240000, 794841084240299249838540552360,

    5613046479518375769715959868370, 37950683475801290726109681651502,

    246358108815143096058331811635584

                                                                        n
       Theorem Number , 286,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

200, 381378, 109966464, 12785669928, 843100031630, 37308611795590,

    1222093335559986, 31565869343206738, 671574552005520000,

    12147063892369171720, 191290217960830428190, 2671661219900276081000,

    33582331642247509026816, 384455335761316483443654,

    4048012717866491168369160, 39523453960683579481554910,

    360324100444326813037839642, 3085527057388189809025286016,

    24944975376457333127740525352, 191243436615739440809470162610,

    1395833765253503879396880787310, 9732394090324408518790410240000,

    65023761729873440752242763549696, 417422828369982304110100799507894

                                                                        n
       Theorem Number , 287,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 4](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

726, 926810, 226816110, 23959993000, 1480458727296, 62447563493512,

    1970779962800000, 49397574639798890, 1025094668897208456,

    18154215148201903744, 280740184208429731714, 3859207398142899325670,

    47833936160738488320000, 540801139717502698865338,

    5630497514325905093438056, 54416840770889418425778334,

    491515508739405829189732370, 4173263281048085306657880840,

    33475290750406690116909103298, 254787563991266470709164327680,

    1847143996791483692235537995184, 12798567745372122037774988022830,

    85009198710504638108573714150280, 542726630389911374663306719728584

                                                                        n
       Theorem Number , 288,  Let , c[7, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 4](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2399, 2184192, 460160000, 44421120000, 2579713920000, 103913565440000,

    3163315728651264, 77006556979226389, 1559674329854080000,

    27057363396998400000, 411031811705904000000, 5562874904214707078144,

    68006089297905275520000, 759451353194964055680000,

    7819750616150190858407930, 74819157116810382466560000,

    669627735463557056561338368, 5637918525894721311500160000,

    44874623936024130964417920000, 339108740120990326562455680000,

    2442118950919896534070198400000, 16816231366911057269236333443979,

    111047592996741791834102747520000, 705109902830308887030969674240000

                                                                        n
       Theorem Number , 289,  Let , c[7, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



             Then , 1/11 c[7, 5](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

187295, 1847388088, 2512614464640, 1186784587021368, 281590223512574080,

    40712238650516753992, 4029256906990409282576, 294715902786589153944176,

    16810706483993626218195088, 777820785095275658602504960,

    30080099328661178430942418704, 995256540520514744010156798380,

    28704924840446902316123320605296, 732733202061689764305891155216384,

    16763511274242733482430961687575680, 347370890019359388192216042733144888,

    6578346224923640105225707129039890448,

    114727003142405973750720295934698105744,

    1854903512834633810475973302393783427952,

    27963795475780220521119047192959802302720,

    395090563938428951532824664821622908849424,

    5255011065095401851714700793172837121997712,

    66063506204811269095927356241093149518378125,

    787791409188330473042017084439211407658832096

                                                                        n
       Theorem Number , 290,  Let , c[7, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 6](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

78, 541934, 349686066, 76860972206, 8738011820324, 627741398479616,

    31969654787604480, 1242114081596814546, 38717489386082962816,

    1004034772050180355044, 22259780937612841077838, 430900868953083925765120,

    7405838579043293692493454, 114546192136493981615470208,

    1612222943387582510903915158, 20841913075179367007421361134,

    249415662557061338267616753408, 2781587735181859177958405493020,

    29077269696376872769541553537024, 286344122479893509111400180878236,

    2668139982933929716030546506146738, 23615721525841793221336962337889920,

    199233781819720338549935073239508046, 1607051336941062890819812270043989440

                                                                        n
       Theorem Number , 291,  Let , c[7, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 6](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

366, 1501916, 796036690, 156413966194, 16495835938514, 1121737906424690,

    54752014496002560, 2055845348853252096, 62296723393174479872,

    1577361804838675714574, 34259234505129918918610, 651398697149371304622244,

    11019715033879136850704210, 168054565190927736680839886,

    2335544902296585201254205760, 29848018827634885602094366322,

    353475989282203551949936244636, 3904525196220313061328911486158,

    40457459463359525470686685736146, 395176651466248583387000039997440,

    3654462020761967742181680719941870, 32118304998141327055780156245789368,

    269185110792157028152478499362684526, 2157911216829764159825599890868136960

                                                                        n
       Theorem Number , 292,  Let , c[7, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5604, 10352526, 3896644480, 624448251392, 57271741693952, 3511354005176156,

    158074981543404434, 5559353743817922574, 159536521896768584494,

    3857078520134092419662, 80497996169858320012278, 1478099234107810827850702,

    24245131742426835642542254, 359696163380859624568727040,

    4876391817526860041250539776, 60933636520810859993024555520,

    706953168953827617690024579054, 7663490901046285906374641172594,

    78041569047473640350498342707026, 750151069097213360112263257391104,

    6834477440885256656320288986258404, 59237552182181535720257197860104370,

    490059492952918579977843286023536640, 3880919895622230103033150624276971520

                                                                        n
       Theorem Number , 293,  Let , c[7, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[7, 8](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2809, 14154341, 10549941398, 2935313422484, 433387444962577, 40668416024499554,

    2700921296630379678, 136213892515853807091, 5479355424841536463719,

    182241785729752590745551, 5150072742206337295463160,

    126315976369166755472467687, 2734967562037112650551263795,

    53001813849552086338321106944, 929888802961216664644660161120,

    14911732557535875681784774144584, 220346625514250560401394629988864,

    3021249037219503470878648608446420, 38670611503078694415484208237830678,

    464481024911579374098139178375034379, 5259544017086740301428733734144426859,

    56375460299881568728043252600230409903,

    574072572416449276537431089396344004255,

    5571655714863199530467995188963969266381

                                                                        n
       Theorem Number , 294,  Let , c[7, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[7, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

29, 89040, 35120527, 5326896816, 446678617624, 24708123771936, 998762632915419,

    31509559987886160, 811858297565166072, 17654517941403268176,

    332140567579290316404, 5510879015848275587792, 81881016214647841797234,

    1103023192370262875792832, 13609957785326731695911621,

    155133381893369883394577888, 1645382233733027932985920331,

    16338981991054900648662165904, 152720505530796614636276101310,

    1349901089348699240759720641408, 11329458239567254434454075951627,

    90611299613498493560255701336240, 692804299443253225795228364997915,

    5078489055023773544991524749479200

                                                                        n
       Theorem Number , 295,  Let , c[7, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[7, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1082, 1134496, 285789267, 33223027376, 2321914405897, 112196325405072,

    4080530630712293, 118172025935189168, 2835328311406874178,

    58033559570801214048, 1036157248514441595418, 16422260441348401127280,

    234306573748308533359225, 3044017559093012574273440,

    36352979614688916511185201, 402276744808065541613583232,

    4152840944993742577602802206, 40228064092348853672898135424,

    367508269636983008597711246058, 3180333834443898926335036437856,

    26171530701501387021122804013329, 205506940235138358473927355817792,

    1544513299355771126807140260155906, 11140693982509997461692903826919760

                                                                        n
       Theorem Number , 296,  Let , c[7, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



             Then , 1/13 c[7, 9](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

13507895, 1991126161440, 23560910673140114, 71789263280088081984,

    90117912980118666676625, 59666685343603801305606880,

    24192132837852145300430741862, 6629158145873174338244287295712,

    1315665798411092748703679491686052, 198982903909330700448443742156305920,

    23837540319551390684261471035129064985,

    2331374911287681753614877635573713275440,

    190707261931816909162255148574529364373308,

    13307044082993997046335330466242092585581280,

    805075430729452560938466798004808470764033850,

    42812799348154026377467352307660823161943929760,

    2024587167829053880394806505826450665066661316618,

    85990563243148667621748870715517073088923011153280,

    3308719840190875723029684219297236206529443465990645,

    116206311660625705806961197173638311931150262953222400,

    3749982720226832367964930155978636530184386662912876240,

    111839711593670286113784661434722752136457003046827204736,

    3098753543212301310942686654537071455721000960561745879904,

    80134684364703961660035239680685654615995212356378342508384

                                                                        n
       Theorem Number , 297,  Let , c[7, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 7
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



              Then , 1/19 c[7, 9](19 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

86247, 6502137834720, 2540585245300543309, 103716033424677715959104,

    1092852880261511204632316370, 4535739784018487705114512360224,

    9416389802321454444502316879408395, 11372278857757324999365183976654111264,

    8854323628393525356052601496754551204020,

    4784301511071023130556096583223486231429120,

    1895565468679029768743617684516352523132031537,

    574553105460347186733339586778784567660044025600,

    137765426952088064857560856029381736636040275021445,

    26847390076126908801300415045020223188574579224849184,

    4347545134494470905424884204536557635745514001821108520,

    595915604412070098475352710469195096581720163320040966880,

    70224198615899830394228587616757411017004232292436742995925,

    7209697711532650819862471883020287075422643374879379815755536,

    652290334149620428707340034758588430874332012700969280995593634,

    52525537147781143482112955830644167762268745895914857646792502240,

    3797336379973792030054545787349615202158592407258253087111289028535,

    248365479553408213630896222317029661594650025302209874706914090949280,

    14796340796517377001408755993959711987568745780494310700960529656714354,

    807785208933037351648713377145352965627369596416456731711233744009445376

                                                                        n
      Theorem Number , 298,  Let , c[7, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 11](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

26, 1871318, 5470733171, 4094607869884, 1349890039481044, 252586657684097216,

    30970343127555864240, 2725502590844018901964, 183185798502460744826818,

    9833064498064378108369290, 435877377117711929227334775,

    16373229062581287960500059088, 531958953899316590482966392810,

    15197443370150134374961953132742, 386998791813665220679548098012438,

    8884132555780752556061216581775276, 185627457263395403129707701866647898,

    3559094791753465736502770275343797430,

    63061781295606791518912113210515484999,

    1038920039908353279920953823251047614964,

    15999906008295071822134435986212987271672,

    231434145219854529005254265499670686120564,

    3157446667433817725610182996824613495759905,

    40782209136578813086499986532289739347777412

                                                                        n
      Theorem Number , 299,  Let , c[7, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1181, 22262908, 40310407948, 23007116456386, 6327803483507782,

    1037118780546332552, 114777078040746228570, 9301577062041676165402,

    583946973750101085226396, 29587405458590021913885772,

    1247987477518771797679825882, 44890139328260457899300418948,

    1403678339977026455152476653158, 38755496857678852177906982936556,

    957062900877006506978487529879200, 21368374469044761689514784636152938,

    435305593171404656151637574709855265, 8154676874922190322356310293000285144,

    141431433631593401071586658129968051168,

    2284394560487538519845561898206108564998,

    34540522094189469071080678827764522134899,

    491139001560183210607084973955890129740604,

    6594201734862851417482607349130067130460744,

    83903339015559505941836921370948589162178028

                                                                        n
      Theorem Number , 300,  Let , c[7, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 11](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6222, 71791187, 105670022688, 53333186271674, 13484468741252914,

    2076339122498513918, 218851875317049342032, 17049151975198257957972,

    1035744637071931968334340, 51035005112541198566845973,

    2101394053817324363684638596, 74009955398720911301984277228,

    2271441775074524417774630750028, 61677394684846310797165536681885,

    1500414597966067024772382646469480, 33046625863304178627685130678295240,

    664893636306417345863978699635740178,

    12314380908886364920516215320631707794,

    211342438112355113725126008834178477828,

    3380536238773082351951068152778849347092,

    50654220107877186744362786880693075478892,

    714213660719582981597121556995387108378753,

    9513909776989610250133911837128676525608404,

    120160217750604409297633950544187406294661026

                                                                        n
      Theorem Number , 301,  Let , c[7, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 11](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

29278, 222764596, 271222088127, 121932861815120, 28449279709037022,

    4125114480506792510, 414745590160078125347, 31092445557259608608472,

    1829291188141444221972794, 87708710240230778042264568,

    3527144653415396510826059594, 121677976117770546648106664936,

    3666502953459628044489915326974, 97936800316494127696928001651488,

    2347481683822964233226528557308040, 51013112424196228229750753979673328,

    1013857396027479355697210979761206266,

    18567087630332222759037564153339955292,

    315358109087255756201347609943384988115,

    4995989777896468110451900939411745819348,

    74193112858594347628380749508490729088980,

    1037405341000940033302776198649685884815216,

    13711471473235480404367505536598393426120354,

    171909090442215277914947084097646089374440408

                                                                        n
      Theorem Number , 302,  Let , c[7, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[7, 12](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

452807, 134375826886, 2263281499655509, 8794864347706657661,

    13359483966033085085499, 10381521697865760533558304,

    4842813005880370471264669209, 1505455554612293350001250042863,

    335416310957268299996087256404405, 56485814610323259398183826312336173,

    7485550528547736409142973897195766486,

    805504262960701179380801611950281772338,

    72167813999341267912135388003732935394912,

    5494021868960839365727370773465426056717270,

    361420294614241711159110121990502850299004309,

    20836722888807590513089701960073885573739617199,

    1065440818789947950740238588626089579174138458948,

    48815436835132576881294965186318507491526541426985,

    2021883062271333892642522926625182404188132949051349,

    76291430689501745589193018959711238192624973233427099,

    2640326340049015290195229876179735223922281743001626623,

    84314251478010299712833840785187409005953612889144488434,

    2497564996692959618048786145588580322039984413348254271982,

    68955751862494041187447983517515439674431052966783601281477

                                                                        n
      Theorem Number , 303,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 13](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

260, 15031660, 52148436740, 48964885177340, 20458692220260936,

    4843596732749647386, 747354739469640584756, 82225398771905555376555,

    6863011910897789140026248, 454519821112648376993159023,

    24705860547044210869135737456, 1131470548255210674140180152112,

    44579107955733979499527447026228, 1536736901945348495913953897129777,

    46999573794227219431983502468504072, 1290237584144106849510080896423272008,

    32107418124853750172658667718903528928,

    730400615794982335492588758892528091968,

    15300213934525822325223394508645419381272,

    297006849790315958978995279306288775625697,

    5372569178122916765700118225146618025955476,

    91007215626909468320675489353282839938039484,

    1449911849380242236724458194152682546996312560,

    21810760979692742247662958806433887236878054249

                                                                        n
      Theorem Number , 304,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 13](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1727, 54004324, 147775906773, 120970006091052, 46070287442213706,

    10187996783790376016, 1490832870805586858695, 157156584791150725966380,

    12659848656345305338980568, 813583098934581038757151784,

    43090917590992932232400241870, 1929230619301929459128741517808,

    74502106946398154669658497255847, 2522711476705464197549492492753836,

    75922403862312144258918491795480182, 2054036946775401377727362625423629264,

    50438722860972121388574454708963286431,

    1133495178598885439316400000052955226820,

    23478600954154875853156133908567152383621,

    451046638000514299001104654449638182207628,

    8080494551505336976431604775329817145586543,

    135649087099927751014724034934644892796518452,

    2142994457629029714511364100830710527511476059,

    31982732796174381783583465747138799631011014796

                                                                        n
      Theorem Number , 305,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

50186, 612271592, 1107023934016, 705773572736396, 226166689818982710,

    43971237349583596516, 5816361211701346827522, 564844445862016097803184,

    42492897978152903368679790, 2576371094111339017418064020,

    129756043051333090570076282508, 5558595284615487070372565080332,

    206429850950871144478976371686950, 6749775925827334752871886913206292,

    196836776361492663029913343622023422, 5175150343505215497214126656461402452,

    123804460649219663853483368882470196560,

    2716311192540873785475937318408899326764,

    55033649054341155579655100503549245612077,

    1035814772293730974087770271958771861317136,

    18206556487687571345955732166372118047378713,

    300255003694867301134384016127669502798722152,

    4665224063870536384759613372787230604482926776,

    68547029794815154539148272733759554367248784464

                                                                        n
      Theorem Number , 306,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[7, 13](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

252111832, 3698415423610353, 1216441736489323605080, 57707136646048862288212391,

    794389078536088600817717095568, 4530254724611220517074945960911973,

    13220701580822132424035922001901453088,

    22660914316493417003387394324990865840267,

    25109658368532823543275282326095088156390576,

    19290164380255229206755971721904031887876589194,

    10833232973822383784735562602049820127665632290608,

    4634604010994358411863709466729484778191385806869733,

    1560881328463493961914721359147810529035838802436932028,

    425034653888212968617225410013566885902789945619962471306,

    95664440099420377102730797047053908831445854087559383902480,

    18128641574373357119327914495410119174981529476766128283609366,

    2938068127205038724181128255347134794483877042327298198488838240,

    412721782843581341822268943804097358267725617892299523360818594560,

    50836890288299952288817408303247283254708314290931155958111916120120,

    5546342729251899675957698125899989440766045765923048012661630485549759,

    540735438330556628766835307097666881061545345078532598433240965664568336,

    47479874103377972285994153852411527938493084423133806316009284898311639000,
    378093369292000659577269760014726658512582809553568466926748390592812217\

    4224, 2747578219989688893857334522310608190945298644252442259436196346015\
    89967856092

                                                                        n
      Theorem Number , 307,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[7, 13](19 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3636, 18039694539020, 67770091970046056868, 15655278202477385451618870,

    710769175532289800731641611824, 10695623001709506955477375821118671,

    71271054146479554982736969169868541060,

    252001017597235003107309823001814811617303,

    534178049771668192477431966266777780472325680,

    740584461080742586467981261161154428920541102575,

    716433217882685163267257472662541395644171205993608,

    508240382834457239650633727950333141406782289775219105,

    274951831986301389859934137490969205702448904522114934956,

    117063016202708852190359030970690356049244285661701169552158,

    40249919414326718281854638907945927132486910066590233850485364,

    11418540751481589744804440055690922853231282848566451874757906659,

    2721478186694377812953527718280027676938600243656467055009460958384,

    553391781572330844676222891746010483805168331535699953551483420630176,

    97284644478894675581902151276302179753399385623212782926262147546353648,

    14956460389130644352248701215296191214467169860191637739186715007975852388,
    203118192610487091458683397459703246544337221282007680319605369101252383\

    5284, 2458359459990774273451299309435807328740530365910079916337649362224\

    75234397556, 267252438568654534343389248918051847773150628662099087763563\

    64392230626227109832, 262790685790760764367785870270465955136298937020193\
    4463382887355565569345028543913

                                                                        n
      Theorem Number , 308,  Let , c[7, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[7, 13](23 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

15274, 1185852626118984, 25025121365362560331995, 22795853621461741389314534048,

    3348449144277858643055394840674067, 143002113733343988229896278651029010448,

    2459280950458146036305271018578640000830687,

    20862139502769829642349474310274646981336586576,

    100083983632435606248120474010172562389276641849403,

    299305863166356067291720829832996426688123183569713960,

    599812940469116113364928413802288234694255121176590537196,

    851445297294185833428036051795176552268788485427357613329064,

    894341949813287604981295786928440885357266085025701350606370763,

    719960672144942088603697829841441208368496470181196133874273050904,

    457149732603229828210792823616269530099921665181665179020790651824928,

    234492875906149106574377115570322900385425165799833150149582351204176184,

    99142242551875256786352365759158049567593436071337561172421444442056643751,
    351466650142573238719568265276599744179325534932625255273087304162698938\

    88108, 106023409375198873834288285410283769366036823566485602245905487956\

    02599808151381, 275650893107324611301417074962716356307692006026255599779\

    8622998706413626449199392, 6246065091511321992233212506699124689779037610\

    61238503681640350157884190546358921689, 124569501850600389672386568551650\

    480271440573685877653927997687040296580440463799434736, 22057489896850879\

    775821339162197728985015309380637480772668615511568883195735669538727323,
    349468261600058958499703516172426276787154350716143239867032961547724432\
    7338061868463490528

                                                                        n
      Theorem Number , 309,  Let , c[7, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[7, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

49, 432469, 404410699, 130547263112, 21667453455795, 2248680582617462,

    163562912227526427, 8976112674485585041, 391095757328808086611,

    14040055207395331776102, 427069329529765327503973,

    11249173305803880359649716, 261071262521870669703174767,

    5414063475468732930663671917, 101497955628062637567250395425,

    1736936326732163681439838108681, 27357847522938640391245401235095,

    399407657350409910811661658603298, 5437969364253864631589627094422923,

    69414820814755211366792572164556302, 834622593635379648630417916676478797,

    9491663193975185231888910518291792723,

    102471178104211107226597049555620829605,

    1053640047914506486794030153659726785753

                                                                        n
      Theorem Number , 310,  Let , c[7, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[7, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2884, 7912023, 4498279600, 1077855875502, 145687836839041, 12974104261519877,

    836989060542692498, 41666036171887880648, 1673623197289123238319,

    56062910208328974645633, 1606164983655927918193565,

    40142514338535442339838864, 889255110745632087189140275,

    17689022953755218808198997730, 319394590836969606451192242739,

    5282561607081929639963228468244, 80651891690025064804361901438576,

    1144263342584513251603040925619533, 15173446118766997667263624153233949,

    189007700235555465392451589541871841, 2221467625972014662420076761415029015,

    24732839149532536284877669000288447350,

    261760173695714777984817768012779755963,

    2641752327118682441567591234909515869518

                                                                        n
      Theorem Number , 311,  Let , c[7, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[7, 15](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

15004, 441465304, 1412591598800, 1411754771576296, 662970569232566438,

    180773348701136726829, 32499930139943124344192, 4188622731095940483684416,

    410362088542008477099920744, 31905282428842258829786205224,

    2034092664977734846001055147136, 109095296279684665399259461512084,

    5024111238459618031826617236982816, 202012221820052725153914283131232235,

    7190569196257373485896349755899885792,

    229221200643208291155303009973723207456,

    6608942611945238652212430230813787386084,

    173807094742011846269239862763645205290024,

    4199924811023862414494227204495659738108424,

    93849057704763676334129998805183518407689744,

    1950172881315046451588502749595885837542683392,

    37873063640426534294421803156540459070158770350,

    690436391809417224382652766182576535711898130432,

    11862391734539080470528182845530750227231755027232

                                                                        n
      Theorem Number , 312,  Let , c[7, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[7, 15](19 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

568141080, 105254557984755175, 199492992787270802040864,

    39268131029921355288979644960, 1850988351011438222251911613625248,

    31743536497586937863129916901024271216,

    253239784750566119631614052130506668390400,

    1102062518151196599263013204412465025731705784,

    2921440724736260867702426886401300801528392207400,

    5111534327516468850519535575816732867424907675646593,

    6271412399694451994935158277401788313098417668615777536,

    5655508908092375718067485384411835329518511150774064560192,

    3891582302274904472677508333109562009841785625281348420640018,

    2106287138893608550822997898735380771412008435203409795143343904,

    919439790230772243577598624725509650359401846470193971458600623488,

    330553344911449992694332708759503877432953036979621767322608482700008,

    99626524961891933032173738252544825134414008904240193728839834960402296,

    25557239697374817731597496795020524152927012244056637290826215504796994322,
    565392233549806656618659538108920094340677535783396101397492718061417163\

    9680, 1091034236966866794788272170590366500731819076460391839284917520635\

    183278302496, 18549260165115707057444856316181346568376477212880742745917\

    5692016290180781390936, 2803192202761741306201694051075877016960159269705\

    8036541270503054055753647172185616, 3795131488336119098658935611240194633\

    687352847304605379992784916433394071031297947392, 46354683471680342934249\
    5150733305957318283148730757863407371314536027539704856527882160

                                                                        n
      Theorem Number , 313,  Let , c[7, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[7, 16](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

28742644, 13352029566237, 430024575556679647, 3229460660249567900093,

    9291191495136748328780777, 13325625939533918882647630673,

    11183084749310824520709746678930, 6107512722046282066354022270702319,

    2339574538441221092870667533024971861,

    664291134682497333281212365751397568699,

    145819716663441378185642627622622591632486,

    25576563716046519416257140378246980712020307,

    3680619704613295506490523892037584872479219178,

    444045160252251283592200884565571006159881892132,

    45722859711633090290495019292497442224783499853538,

    4079159354458960703666073572464364956827655839391924,

    319367370453647652357380230527433178645284391102469867,

    22185025925352102801754369457690562384552898882525019886,

    1380412129858548838009292095929512437047141527617892993306,

    77578876510020311143790351148680249424854804803176537565491,

    3966759582518924687604197930630595157194624422933614565060503,

    185734806386094281789827346868461836050564623087037001174911649,

    8009675450410237906957344573697189435643097438883245489970304669,

    319769687729220415391326656557156903173371807389349786144494245251

                                                                        n
      Theorem Number , 314,  Let , c[7, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 7
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[7, 17](23 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

19269350, 1137677075614864208, 60103895783632909465452361,

    148424552392101830718733370192992, 57679904972860256411826236257607892607,

    6249425749930895607589219349177935412304800,

    261256592081402571374407277996358599341616005463,

    5178246138143542942828470876208369281432774845398416,

    56004641898168378335493182496573062329644758500357878649,

    365655022570673720757231793064665362642030578195181151317680,

    1554336587952680305676486437090127802940122132495217524065720575,

    4559996867709846276142599621609292000754379028845761724052798153680,

    9668458862149208013422736892779966128777464251446270845500197065140345,

    15377744007101471772371325168883878917187218111555447860770356658427582616,
    189169763302189834373366024061308409719983821726904512114651337165961762\

    32097, 184630627140742653925969122998259934335047784304176643422816944021\

    86759980666368, 146080147222910774436531202930654689540680026201441129680\

    08374922630918154444569335, 954302300655262963811202310232903672917886410\

    3561770721423159529488124274862484983392, 5229450952109978514383875358735\

    297448017473656877179968977783370767451252602496062063121, 24370762666094\
    962010481144727631979125113934024322867265796315507961120993621749993401\

    00032, 977574009060850164410214489753906521114737546950464731481964077868\

    505195689333749765695132700, 34111673455369745072352918728263345366492806\

    1960569005653693562976717667630890102487057517006240, 1045234533764136244\
    880471098393943068898533963224417794715369874120969928312463432512201852\

    91140606, 283611980533137547561077242868287023709740736566653574335577783\
    46140300460041147083486331109107089136

                                                                        n
       Theorem Number , 315,  Let , c[8, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[8, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

19, 1024, 23040, 327870, 3468800, 29666304, 215644731, 1376931840, 7902996480,

    41467581070, 201478531584, 915582302208, 3922705138950, 15948484346880,

    61863738831360, 229983185651712, 822550281176576, 2839626634798080,

    9489123960598625, 30770393654528000, 97034737429089792, 298156578116505390,

    894185894748743680, 2621453058626964480

                                                                        n
       Theorem Number , 316,  Let , c[8, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 2](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 1855, 196722, 8880628, 243205956, 4741797888, 71938576420, 897778600884,

    9569090608128, 89475348746900, 748726091702244, 5693474376359175,

    39820433246110114, 258658177671502164, 1572871835176178688,

    9013239633554081332, 48945155664259126150, 253070743662073350144,

    1250983796530598779364, 5933055864188357830656, 27081235420728828878178,

    119290885402871684551390, 508329382566321951265188,

    2099995343637338394058752

                                                                        n
       Theorem Number , 317,  Let , c[8, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

50, 13824, 973858, 35042762, 826159104, 14449206074, 201640307712,

    2353623083370, 23740468153842, 211898196038362, 1703775980878848,

    12513306812580234, 84878705008954368, 536511536849246208,

    3183591627550741698, 17844025065894406954, 94967068493735089842,

    482056078159441217754, 2342831256818740530850, 10938585683317383793152,

    49208092716775330206720, 213843355501608686541946, 899790213925137484084224,

    3673399564366593347279360

                                                                        n
       Theorem Number , 318,  Let , c[8, 2](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 2
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

186, 34882, 2081280, 67855360, 1495412378, 24880548642, 333957351178,

    3777335170066, 37118243298816, 324039912137090, 2556123263410176,

    18462236192716800, 123394359343049626, 769749852643599458,

    4513730106369641472, 25028922200348209490, 131906237369365953914,

    663570747418277099520, 3198420929803658421396, 14819369241307096674994,

    66193585376485249218048, 285756341925866207028224,

    1194948516542872496164490, 4850108722087007037031314

                                                                        n
       Theorem Number , 319,  Let , c[8, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[8, 3](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30, 2500316, 5320732755, 2715105678390, 603407848963792, 76366891380104446,

    6384574199184705938, 386791492369342497110, 18072704935672635977740,

    680869260806422705141880, 21375946897731372248176228,

    573583641411413424716139126, 13419235928699860153010398112,

    278146563415516795336021987207, 5175202709958605240953759733664,

    87381437035124077363452181188504, 1351235376883125000944918654296040,

    19286459664899885954799659237456724, 255798417840374563098877669839358902,

    3170985819076061252444608860875238040,

    36927565659275775844157373675436722220,

    405799003142444777733313199714483511444,

    4224709452959636430761112159457091142308,

    41815927297340868533652615026545408244428

                                                                        n
       Theorem Number , 320,  Let , c[8, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

367, 131072, 13303808, 690749440, 23174873088, 567678079284, 10928002498560,

    173541817057280, 2352171438604288, 27898553392955392, 295112587605996942,

    2825739695807266816, 24783805920189087744, 201045094505369698304,

    1520489832754140545024, 10793284420094178516952, 72323391317430867755008,

    459707612851179267293184, 2783552511757719208067072,

    16115159806522770904580096, 89494333400203869444163237,

    478109760327933381923831808, 2463421427201018509794148352,

    12269295364690894300305948672

                                                                        n
       Theorem Number , 321,  Let , c[8, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[8, 5](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

32, 3328, 130176, 3022240, 49904192, 644254336, 6885055232, 63252387904,

    512956587616, 3744421948832, 24970977630592, 153914199571808,

    885021516725184, 4783613158717632, 24457711278496512, 118912193857565632,

    552250933261421344, 2459341374963181280, 10537131446002554656,

    43562447441992000544, 174221149531187775296, 675573680990052393664,

    2545096072494908123456, 9332101969992032378048

                                                                        n
       Theorem Number , 322,  Let , c[8, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[8, 5](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

224, 32703311, 139897251168, 135704070910080, 54372807770416320,

    11880667484179636512, 1655302963474183125888, 162285634483991601097440,

    11970302030203776584453856, 696898204572579015417745440,

    33189434081911739478361622688, 1329072602895136012051341160128,

    45735351812405678434553890384643, 1376309654322121828261217646760608,

    36743471426274935439312033504834624, 880720139395828040182586908187958144,

    19145574222131869842257940143088903840,

    380719112057898584277440251710339287520,

    6976863835191385053382671782356184425504,

    118583738055266640167187350813117530446336,

    1879924226626027252609666008353197625966688,

    27935484090657951423773005951443164183017248,

    390821761405144415218289558299657001828119008,

    5167814832692276826901029076118627369695768416

                                                                        n
       Theorem Number , 323,  Let , c[8, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[8, 6](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 90642, 113423458, 37548412672, 5764730445886, 526441487116078,

    32796714598400000, 1517911588959842382, 55275138223106215424,

    1649815760682284519022, 41605292633064700190238, 907370893421010378281358,

    17430491268445477276122302, 299316330601677174563394560,

    4650440077045868348235200000, 66033196169253501048531030016,

    864188473034560759273881600000, 10499313599453971711537971200000,

    119154481375640950083978263812962, 1269964091437468013187872968922098,

    12771601479871426635456251234575678, 121694064625356782268428501046167762,

    1102696296355173975742656958648028382,

    9532871725622245086135784371734936832

                                                                        n
       Theorem Number , 324,  Let , c[8, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[8, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1982, 6642432, 3505926818, 718097989838, 80209204756478, 5812665446401316,

    303038316534049792, 12158889587507200000, 393532673746997605826,

    10633357225002003306496, 246181416438900507200000,

    4983502145705470121885440, 89644952265263706089883262,

    1451889272257146177294962322, 21402808876402172443243943906,

    289791598390041160503300296846, 3631829936544494873581830457344,

    42408894267754748291655299454034, 464043486490933180569223179861922,

    4781794691864272359752660518155214, 46606898662523286576509083784688382,

    431332765241506743010804599985470446, 3803359997094200556108598860185600000,

    32051368891837165318499621465806624014

                                                                        n
       Theorem Number , 325,  Let , c[8, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[8, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7666, 17625918, 7860539858, 1454243512542, 151420637904818, 10414183790400000,

    521152662379693678, 20227260391278843166, 636812503974818780416,

    16807043191536929164354, 381290476777460531200000,

    7582522058930055951317854, 134267142858103501619200000,

    2144224070492537942940599296, 31210927618525726182849380114,

    417765575800340822566446541824, 5181074929181234119000579978012,

    59920420895221050753790505835520, 649870461013136309275211332973312,

    6641944059352999739431563197091874, 64245498427135673889002064795623346,

    590359651150109516170953067724800000, 5171097089503681979139738358099200000,

    43306345198076465271736428681615942144

                                                                        n
       Theorem Number , 326,  Let , c[8, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 7](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 15328, 6058464, 840767104, 62616999392, 3041948184205, 107563695299744,

    2966465900626208, 66875427746854752, 1274602516336402304,

    21061437134938079613, 307627878788122375456, 4033137870292701653920,

    48051967811861392824544, 525582083706438224783808,

    5322390475366020780320032, 50258761808396257374252000,

    445248786977553730179130944, 3720148545710979789205505088,

    29448708516991595317669810208, 221746109723793157376945321024,

    1593894052371545914330960679072, 10970683136299135573918879931200,

    72508548949720457739517262150752

                                                                        n
       Theorem Number , 327,  Let , c[8, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 7](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

160, 199328, 47852064, 5011425856, 309271350368, 13096362312832,

    416431591340096, 10544722329692864, 221503771955317120, 3976869910810753536,

    62420322398522253504, 871721143108065557664, 10984758318477713929408,

    126333712580196988898880, 1338620540695103520271488,

    13171484693392104979207328, 121159855047401330599889376,

    1047902112502173447909163296, 8563960258338682789372277440,

    66419898896421076339686492096, 490727259680771915559617872800,

    3465435510353631391320905528576, 23461012531325955583859930491328,

    152674004865841800599108197765024

                                                                        n
       Theorem Number , 328,  Let , c[8, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 7](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

832, 653536, 127955488, 11850283264, 671839495648, 26706675919104,

    808162012018112, 19657365535903808, 399315712285642272, 6967826285870944064,

    106704005027835315040, 1458338594941313074944, 18028892181080416669536,

    203833645461968614168448, 2126796852455345836590464,

    20636433573939663590374496, 187421216099450760899175264,

    1602125064004029317242712096, 12952750236111305416685396736,

    99458968581890376261486445696, 728031278241492085893801776320,

    5096863469817772022856546905952, 34227028232124194907767394225216,

    221045523552829799433935516732576

                                                                        n
       Theorem Number , 329,  Let , c[8, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



                 Then , 1/3 c[8, 8](3 n), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1/3, 320, 35648, 1733744, 51780416, 1107797376, 18499639280, 254501953024,

    2991780567872, 30850520776832, 284568815653248, 2383708422822144,

    18350050155144176, 131074719062455168, 875651137988234752,

    5507261421636044064, 32790570565714352960, 185709308457114820224,

    1004541182157283525760, 5208310264812516902656, 25964199800529933742464,

    124794713204397868812928, 579722672153090711429376,

    2608512208577635354065408

                                                                        n
       Theorem Number , 330,  Let , c[8, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/3 c[8, 8](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48, 8352, 506928, 17417664, 411575424, 7411551840, 108253146528, 1336587774336,

    14362700340768, 137242124727840, 1185420555500592, 9375107780890368,

    68592716532995136, 468205814657601504, 3002540850739717056,

    18196488612050381568, 104738114365714137936, 575051519312501609376,

    3022815704134106334336, 15262772624610469951680, 74236309371208411685376,

    348708491438772413462976, 1585453591604910240807264,

    6991468430311317198570624

                                                                        n
       Theorem Number , 331,  Let , c[8, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[8, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1313, 1610886, 458910971, 59234332002, 4541930609435, 238726670655425,

    9382441260116734, 292092207525074925, 7501537828733691643,

    163757504991451321505, 3108723630600635711279, 52246780582103957096604,

    788601621444450454133542, 10815779551653448486543301,

    136104858533750459661849260, 1584331893126254469591176322,

    17178529442282738648183637276, 174533419058124294307369121569,

    1670184237397788965977267742584, 15121681830422798380363864683655,

    130048979072538016629805162513617, 1066126229292582200882450529760032,

    8357219195678298654178043334595477, 62817510651308059380138590175484853

                                                                        n
       Theorem Number , 332,  Let , c[8, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 8
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



              Then , 1/17 c[8, 9](17 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1927, 247924007466, 85909472801492625, 2865922844964809033367,

    24279733401442795078183641, 81019772811678305329675507501,

    135911248648381970575020981288557, 133499169453525921255244754978314100,

    85114294923317120516755684042457070552,

    37910127059481847583890583516502948395378,

    12458899298924532058383313138441813915088038,

    3150709651595577033018569775318490708707217883,

    633735264530502006670214165225433352048897468935,

    104121446475016445145030438516290145268171153368455,

    14281577590923691263294728340848399258685513830101176,

    1665293735182024004665443051510356251859552273790987982,

    167616629437289934411215807276383685361299765535200847541,

    14753821698968280763120228012322410060986813293551835883915,

    1148451316283482700659603042966450301901255344130536481074065,

    79828418949871232418360488777754533042270851263866838867106294,

    4997190212597315693671718538280517465646175023492040570154878551,

    283833345477129447985223613588812693866911065240581723150404168981,

    14724675190009343145268200653302036721036074855684623123792085413110,

    701834770868672383582780361187220756765851351852074918247031440128459

                                                                        n
      Theorem Number , 333,  Let , c[8, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[8, 11](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

67, 17860, 1566894, 74948749, 2399694129, 57347926518, 1093210304507,

    17374649709847, 237535216304319, 2858706092611738, 30825455872030429,

    301976246640489603, 2717677886578218708, 22674773651136820226,

    176724005881140170283, 1294861500782267369625, 8967734480109139874599,

    58979408920729489189098, 369857728620448061936818,

    2219336776318155628877432, 12782623855814029694221098,

    70863900754154106370625053, 379059885982724933121527285,

    1960769223766964251545425613

                                                                        n
      Theorem Number , 334,  Let , c[8, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[8, 11](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1255184669, 5439786330358311, 835198101392862183161, 21955006293085025549928819,

    183824720814278820308638596594, 676952511461997337845573268838677,

    1330933309136962790667610394968168626,

    1586769538703495300840768412981426158934,

    1254104737384369893269643111548594205883696,

    701407483865106869645975764345602520753618783,

    291704803190170747000468418383553784958604414040,

    93767166099677466567153484937452755681008561069580,

    24027343017347922471806978870123176051094311027462451,

    5032974669346672260094400317293429789254834910865553667,

    879902548174285117962792745334439618113208009119831092535,

    130647823751874530023419544735262899436553688086464529700648,

    16720377526930988439180921397935613166277017210213009441339484,

    1867958095082178506197884071188080503814390695908809506455615793,

    184171652471266181026585061283754663051369785384019769477553685020,

    16179344565474027164212672697468837803017096903212823341038852077084,

    1277097809259785461047408883456412846836380996199123187655532617511330,

    91249864737829950165788031275591664925133661188408490382276886187824475,

    5940839916272279589376913506195524547159262192737646715982544742804625905,

    354511173738334424534702483046809424895233550925369034231027081959534184288

                                                                        n
      Theorem Number , 335,  Let , c[8, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 12](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 67798, 69711944, 22125675429, 3464274876504, 332837164299374,

    22203346773778072, 1112195689914783782, 44130109833966018468,

    1441413818559428183354, 39889334664954813611768, 956331113687410969591099,

    20216328053200952756495960, 382235878061699053302573494,

    6540175419856862785475440376, 102265024159648423080123951638,

    1473455846627727444586648979768, 19701015433831833915677973518538,

    245939843638556105425455958281288, 2881779547399175430257492361794755,

    31842153397407234601624658001181780, 333146109136138747885385633204299818,

    3312368503054626819603092757641111168,

    31399693110240878791991953900113903885

                                                                        n
      Theorem Number , 336,  Let , c[8, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 12](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

360, 1318532, 776005568, 179045629444, 22620838983784, 1856522842768001,

    109590173908816732, 4973902807199476686, 181852535438461736992,

    5541913217803694487455, 144466890372284429396508, 3287227262469813674950098,

    66353376969270972288449336, 1203883596888461689794229977,

    19848359869849468558992831232, 300090382531115130571852660870,

    4193107227361981668725308614144, 54508735678427773431556344728888,

    663046231587091443724244598385508, 7584931194276476274595537723146352,

    81961108558275795054815647778725216, 839864396100942405239627940866246918,

    8189652719389094809691978588464432052,

    76230245343296528196273668564886499122

                                                                        n
      Theorem Number , 337,  Let , c[8, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 12](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2373, 5230012, 2447576322, 491127838160, 56314421922929, 4298062006004292,

    239630451538910562, 10381887970950907056, 365116225503856926434,

    10764570674056970992700, 272682849002704438436632,

    6050612046503711081673552, 119439600305354677707857053,

    2124233436831789434092748928, 34397236343938166596259744978,

    511622473747715167530238250112, 7042796753867848322428492404940,

    90305831521926318265307649976240, 1084661136162608525240166752947526,

    12263255234193118133549187525204784, 131075543178198026020542123866977376,

    1329531657264799569894258185238287892,

    12841410406686639674486457137540645910,

    118463717444985844073274795953206360704

                                                                        n
      Theorem Number , 338,  Let , c[8, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[8, 13](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 1300267, 9110239453, 13068878108438, 7398687444725025, 2221614385232069906,

    417059159484345137765, 54260479886753716406215, 5245875452885217146344281,

    396157279800214678024784912, 24253885577201868156454396125,

    1238757023164728636104188928269, 53987526448490203990780061816903,

    2044625777897326207681298748985879, 68302758036274258861193809907661106,

    2037851005527582150435100739869691938,

    54875223744554121956525038612728635452,

    1345671841357329924234696834713943253775,

    30283824896333812427137739763139821330273,

    629652430577502941448044160843600390874558,

    12166254002368890494175079993223366117814315,

    219595704352726897124532302766838188629817387,

    3719571284129530273756287352390157165919724074,

    59366272703675495213359602199085279699410384547

                                                                        n
      Theorem Number , 339,  Let , c[8, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[8, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11926, 265460646, 666481486954, 535412594175568, 205685609806382281,

    46506460256551013979, 7009271506217361123623, 764201771701893864643487,

    63824785668970913208109979, 4258471510021984146832935864,

    234345579710697914540060262962, 10904987731094404010443855086990,

    437737977705568400025987296759248, 15405479293275968726419024492382655,

    481775823283740567049878421825124167,

    13540084461933996622901408196595768704,

    345273116053650526104367497292653583587,

    8054458300848622273874630724057407602783,

    173113077846583903211743989447208635325406,

    3449354097831208580658207785406510029193487,

    64066074214851870672536455053069146799369660,

    1114539923185941789916090649191834691997230097,

    18239131988125196019882319048861945241585184080,

    281851867188270843425936514282838745521493493249

                                                                        n
      Theorem Number , 340,  Let , c[8, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[8, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

62375, 894053966, 1834552638848, 1302701152991505, 459295018785298101,

    97383097910073581011, 13953647305669757922529, 1459994371807552325332604,

    117816497467299821268790129, 7634013669874834954523206944,

    409587227100846098097280205384, 18640560654320671674494023592894,

    733646849486280496630967285441985, 25368093985994641698834447954485201,

    780816991169335937864329867242644673,

    21629807690076532331754353642813810796,

    544332080999406943372645403147922394806,

    12545142530773664462824476774613585491613,

    266633364043109147669599770061467645374946,

    5258049257711801720407587697444008782791418,

    96723528916001486651462115328654172963506221,

    1667617034648011701161510431289156689537763266,

    27061447799521983276245020456252934161189965691,

    414894582123681264822928597075064441381856273361

                                                                        n
      Theorem Number , 341,  Let , c[8, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[8, 14](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

89, 34210, 4129527, 263483646, 11001799215, 336932264746, 8115363868018,

    161062934983112, 2722343151944717, 40157340813273240, 526734267519353664,

    6234742685084534510, 67388965717696985194, 671604677825259030770,

    6221500349257703421310, 53936573609982542895108, 440141626520477869025276,

    3397708429005065658034394, 24919705960395178792723770,

    174303705367699527627187920, 1166609682501694837409275198,

    7493487481903753311710745160, 46315566613188725786672878345,

    276109258018074949213467224046

                                                                        n
      Theorem Number , 342,  Let , c[8, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3359, 10483352, 6601079529, 1726085749162, 252130498195546, 24094404487963944,

    1658990039133242610, 87763116846983951616, 3732900209554622052786,

    132014181064756191524392, 3982674398249361236276464,

    104582223220619716576312752, 2429369813105946976519855791,

    50585399108410660379479686862, 954600223062124846859344530804,

    16477660537118511929646315678648, 262218150281512104908657105969314,

    3873106669298760103909098161844792, 53411215005705293539713694257873001,

    691206900530892713857624619841726000, 8432302484576421335073893661162710532,

    97360566405101280649520692637914831752,

    1067740811053133998094164297212022514951,

    11157837108584340063925955404762142662938

                                                                        n
      Theorem Number , 343,  Let , c[8, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[8, 14](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

214, 3000490695, 294028221549582, 3795741231813022445, 14709974620904329880484,

    24982871660985218930042896, 22993363231095754717945312210,

    13108428595851624120463826524686, 5068898688340010103057945166971722,

    1418891063729279940226222403774560470,

    301818814012144072008465229078790778108,

    50646545842710892073192214204316040253180,

    6905472645354636605052184605946614184493622,

    783482766438585939020152475016134470914739647,

    75431943486030901697685908544023459352630694744,

    6263966441038434180031950002365947673215824343601,

    454866198269623403336409034166373432247306028263560,

    29224840176093445250491178553897444022569530510780853,

    1678195399233011144607538505489726628537380351554962052,

    86889336538481687796610094667207293060118159007150036431,

    4087527170693960435333329900702938634623587221246612294398,

    175900022458752167961875370933743543566270010122642546809630,

    6966108806185499629715301321579894464254266682060246082413380,

    255246379319829159736093183081816860148981698453126081979509257

                                                                        n
      Theorem Number , 344,  Let , c[8, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[8, 15](23 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

154723, 28746642619663431, 1546854387675318495441649,

    3245216212398975052481532911488, 1013411169080941753350657112208689865,

    86521879232826300107574218656081316258919,

    2833351206344861185899472386316127686398038092,

    43990105131079521399801121426874900883601041623255,

    373697572406376020930340691123920513969074583888993585,

    1924144075752481282562561744403671194106243146503865507058,

    6480008479390558014207973829040792594975935847294103777014580,

    15133565678053908833471302128373769176109259801065673528120297584,

    25666293233155544835287968741335192240530240082745687787234330579681,

    32806970505988099613904309224759453601085604306536141333194187547900951,

    32580897142980961246630656420873506941469675239952137025983769607244192210,
    257836035054611057834027480849348988649509138148661315503013403744073036\

    18326, 166098874107965151365979308109781701024548465104524340412873416506\

    22214430417987, 886993336949896612149102891379824121122185618842969273632\

    2934614252126126810022579, 3988329238093672517735577865897962789044716457\

    447768755298761467317247760262604171787, 15306218540914049185430759812217\

    13153807653668200848014435294018634784044765917262419772, 507342683860772\
    748621675157686912943264679480817133164895401801417263194429171323277992\

    506, 14676644792938396547104209237817037902914788003505755444081396743397\

    7624204395792046327394080, 3739923646220883831017657668046028171277590104\

    0046867755343218951382270892762604887838445461906, 8464285866283566424582\
    553281484652647108509175072928571583826563686955252155822375600382433765\
    775

                                                                        n
      Theorem Number , 345,  Let , c[8, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[8, 16](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2098, 25390624768, 3229686717951680, 56331415902615044096,

    294851685827750728159535, 670284950105016229101010944,

    816858071369163263598979323008, 610058104083175076905329498820136,

    305952640502622282379955192816132350,

    110053745450407257788477587430938296320,

    29830256553987114155743692451943337439424,

    6329531476779404894723738415582543842770944,

    1083591434010451385968991535876163967116652095,

    153370696594044714456467819133983915975132053504,

    18311385400246917714553852956698414595247797399680,

    1875325291786756551616032332782716931561222395396096,

    167091484729435418067342140832850819203757752544822970,

    13110011283652544882027391924867503012235294248574115840,

    915268533143857335603347573447356414088736936417606275684,

    57375442611663040691502131869922088504814707429414059048960,

    3255243500826744615991150994871392965449088816263756633154376,

    168330217047250582423004847383373880077286768690331891104743424,

    7982903189951498939246814606628958709304333051431203533003216640,

    349130961994992849307889235249657256800672695147694948192832716800

                                                                        n
      Theorem Number , 346,  Let , c[8, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[8, 17](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

114, 60435, 9676635, 797755875, 42215861550, 1613971953645, 47938269775926,

    1161309443320425, 23751044522886075, 420730302894535440,

    6583209376059579909, 92406092661533343648, 1178146132442741031675,

    13783668617142470510700, 149240136842042128828515,

    1506160405823022934705395, 14255198647861100559543933,

    127196572772615280146611260, 1074883758276598948367458500,

    8637108817359556472628634245, 66225802871302565404221988734,

    486067162292000278694600448279, 3424421757229809061543308008400,

    23215960489989971717990028388500

                                                                        n
      Theorem Number , 347,  Let , c[8, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 17](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5, 216615, 478653525, 296510449945, 84967370996175, 14250626713731645,

    1599507560817974770, 130844518346750757325, 8270201170285482306420,

    421290245726160395889775, 17852515970006602652033115,

    644930254965959369020475100, 20251910939601452483435788950,

    561561271030318855134145025205, 13929576293993983030794017033100,

    312459658338479776292250893017380, 6396556644005383910059718238775755,

    120447885759084845359700278281067350, 2100350421213667814672320535797469000,

    34117922713946653588694096797391260425,

    518938355625787699096192772106888518130,

    7424596435551361835912215867518663113855,

    100326135070380889644033330102565618320375,

    1285025508675154051711969334161957925100705

                                                                        n
      Theorem Number , 348,  Let , c[8, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 17](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

675, 5805981, 7027037130, 3075772907923, 696785665992255, 98268270269160416,

    9637423297218456636, 706887679465644619005, 40806801503729006481623,

    1924723662812286487669998, 76317943663569168087966756,

    2601257489363017267840643657, 77585788861312997087314806240,

    2054653054337885436925984805040, 48897327131216875641941728182593,

    1056372533072121872012129911148742, 20896298046026896060677082780583208,

    381283872700882439482915604237495624, 6458507926007783903160404473266193638,

    102128372849294901816062469845418266985,

    1515041736041625600816785292178490435338,

    21176554710741151330505839304150050870300,

    279975476525422529010693614960743166346984,

    3513368049091466703055334131793520927611523

                                                                        n
      Theorem Number , 349,  Let , c[8, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[8, 17](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5359, 26822945, 25329516930, 9519800135560, 1938953380938147,

    252438181736721264, 23244254064607675952, 1619414871464385035100,

    89544082105225277297109, 4071061228135325545904272,

    156358782930453187629652933, 5182265290415733883577180547,

    150773008224292053593748571454, 3904893390479758693297143285279,

    91080333545357744335414114041825, 1932059171302856565768763535972067,

    37585168651236280221380044721885625, 675346325419236858951786120461656539,

    11278528842912398380381391114264265828,

    176018062071852283247267853112894796272,

    2579419392608531429335911944005540506765,

    35644374361399883069977184498027721980157,

    466238630390917338344503294021892283239700,

    5792240503650839239969254492115879290534822

                                                                        n
      Theorem Number , 350,  Let , c[8, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 8
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[8, 17](23 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5831075, 858679483833858249, 69231715556463873208130380,

    229646202841765624350463024162770, 112812686005606021542650602631932286550,

    14903010448387551575665581655582037219503895,

    741453553967512899121481666583421998250293555444,

    17185810607753726559543750065472675312929945183380710,

    214475271538639420725063811248901130242985964572135811380,

    1598793134811267820302133069276014687256184768398528174403550,

    7692786310140885753427842021266157755255068083251313809722860345,

    25362560813141602855371470161962320290442968199452231140507302090380,

    60064565580374705313089423268803221810227135175888466489652481885189989,

    106143728071402422310082850204907249884980944126177980050760179275931699520

    , 14440838632087377207444712118388393395106598469448349686507244421794629\

    1779875, 1552446298108226540045007989986002442697868146268023597494353292\

    24384283455521600, 134803852712537866518227260848627037530607619351866570\

    718846891322555095377410125457, 96334579989036012963917449812396953021881\

    403740360150542321050982314680817994335836930, 57578433288770989310019665\

    472585403108472903749659221658108378065570276421045311294184805, 29188932\
    486365507460639510613331337287402945544922043515408464099933220294544109\

    845486685375, 12705190479103376559311174948313339782542654709731296096014\

    681543329278560697099215696095700150, 47999882179010222651260521345827684\

    85444542997815319195488899276785253950492944958620880223536468, 158911328\
    076934955713402471933886617813665601679539916687033294391290547926855749\

    9075843927236119106, 4649798177691133795459775039854356445813855561129992\
    93105454951776836381513169755816278397577165707045

                                                                        n
       Theorem Number , 351,  Let , c[9, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/5 c[9, 1](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48, 11783, 742304, 24094864, 516106528, 8249119664, 105715162182, 1137829654656,

    10620494402784, 87990653972240, 658502073690560, 4512377207673944,

    28619872555717696, 169488959548595648, 943963125676098336,

    4974193539402666688, 24926169946421272288, 119300982650097175376,

    547418408614471009728, 2416023608795083733328, 10285696755908341284960,

    42346270158403181747400, 168974223121463521927904, 654816940230545524079680

                                                                        n
       Theorem Number , 352,  Let , c[9, 1](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i
                                 |  |     1 - q
                                i = 1



               Then , 1/7 c[9, 1](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8, 20752, 4540833, 368647520, 16733094176, 511552649312, 11682721349184,

    212653443453408, 3223126576909960, 41967732404416586, 480468802455118000,

    4923379697919716416, 45793551451575991072, 391013149010336435520,

    3093424801653648431216, 22850016398954449423744, 158614392294543343485376,

    1040401270021591072429632, 6479167838902651653949120,

    38466502839487361809448288, 218500134219580332100207456,

    1191241404163367181869700672, 6250914328213903264913884720,

    31649494019361745516103583760

                                                                        n
       Theorem Number , 353,  Let , c[9, 3](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 3
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[9, 3](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

936, 1368567, 363494352, 40747482000, 2638427817704, 115688582206176,

    3773847478525344, 97365564510561424, 2073076445276010156,

    37572065181650895696, 593361436855885549136, 8315423451096435356208,

    104919354479557409903568, 1205990411182858510849680,

    12751653845048040889566120, 125041044602246939361869607,

    1144961300809673479085807712, 9847795806976398168188208048,

    79966254140124655687694405352, 615769565494521138692026948048,

    4514000700806100341661634497624, 31610242203219488066665961404512,

    212100304826530701473683789889468, 1367366872356712025139789496462128

                                                                        n
       Theorem Number , 354,  Let , c[9, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[9, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

19, 22607, 4025739, 302501584, 13400462976, 410890126479, 9562415114627,

    179143748135936, 2812875562633245, 38111897520742200, 455428249124154832,

    4881657083802269151, 47568495165897753109, 425972261058099617313,

    3536905935317398803453, 27433190478260081915008, 200018179172824486721243,

    1378271340890941836674048, 9017432894957920115440965,

    56242260390139566994886937, 335588052130485172874511944,

    1921611939387272373821655960, 10588633010313876984364301365,

    56285895415921729411975032786

                                                                        n
       Theorem Number , 355,  Let , c[9, 4](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 4
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[9, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

104, 70299, 10081937, 669111296, 27268041191, 786510277949, 17457446038959,

    314822025815136, 4789767416648576, 63188134150420736, 737940644419458217,

    7752911141631521768, 74221985838201018496, 654250045947445541163,

    5355837643984128643944, 41011336875662222981477, 295539954899958599809167,

    2014773911200850145848866, 13052342457446078771155983,

    80668642588751914770038115, 477274990121787600718944329,

    2711435922219179365942130595, 14830943784276362893974758400,

    78293195894778217911016181621

                                                                        n
       Theorem Number , 356,  Let , c[9, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 5](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

162196, 3850014305, 9364465868160, 6997782093265004, 2449948196354110952,

    499146253579143495448, 67344469401793830448232, 6547678397432379984693844,

    486601278330491974876495892, 28857136106911372796885037848,

    1410856138407612604244495570968, 58329743290344912788266887019156,

    2080980914724953237890420835587938, 65127286017130173486744899759365976,

    1812496066102424634779420122587723288,

    45368365977708082448658216317406204248,

    1031288125114607984784663912791456162240,

    21465686908375658085686431089797484073152,

    412047740617590171627217977904823455787416,

    7339886805533240497983910774535470879217104,

    121994693440499752410135990663956581327555112,

    1901063386392583826956895379441109898339580888,

    27894431230896555821082169037561390540046455680,

    386871354168335144196982760416242719246040492546

                                                                        n
       Theorem Number , 357,  Let , c[9, 5](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 5
                                 |  |    (1 - q )
                                i = 1



             Then , 1/23 c[9, 5](23 n + 19), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

832149120, 3346765348952828, 378682504836005484084, 6982442813762952568179903,

    40590162399764534019304894920, 104005871115591730665088808155784,

    143228142651228754773937297921010680,

    120598708150665336120716600804917070024,

    67902120783419759487230046796011073936056,

    27287316406041480021891978221424459965992576,

    8221559066530770090092700290073488527812062456,

    1929661731916642526964089864093052462393721497352,

    363718211178830303321996144105053125616647982373816,

    56433125678503128548634617735324977440382451829071548,

    7355903649048588775055084080338506017357870638150523900,

    819351229472666443548340147436934592506808026069179529224,

    79122399821887242293634615839625449588401872957611861025408,

    6706278511557686515584814496286673898037784783340025263870588,

    504245575156094971125756647004082993716261721418036410709486268,

    33947485930574926770424632258209007736193590988474149686433586240,

    2063047655333729399520137519833292247582086082876896890890406595840,

    113989310299360130379824074126411797926629317788288919075346091065464,

    5762890038117942468945980769317057665516584909332653113843350540331916,

    268107807643718753581358710238493549517108340468221786608198304270602184

                                                                        n
       Theorem Number , 358,  Let , c[9, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[9, 6](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

157, 183168, 41140224, 4042903424, 234852573312, 9387714013491, 282482341101184,

    6783862490702208, 135414105041682432, 2314303889679499264,

    34632193099257834822, 461764623656957902848, 5562649928142821495936,

    61230867875629145205120, 621640719136827174980736,

    5866549290268281081799661, 51805593112070071117725696,

    430511490759974392339070976, 3383268125875935635084822656,

    25251510449333358714112113024, 179666293122716074596078913728,

    1222682254508842432036483634048, 7981929076419220260315540594816,

    50117895995088490802748583870464

                                                                        n
       Theorem Number , 359,  Let , c[9, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[9, 6](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

18, 423218, 496782144, 167751838080, 26974939443858, 2607102222869376,

    172636491835862766, 8506029464875809230, 329831874040684216320,

    10478187595464855417486, 281019241043692353740142,

    6511442444590868149144896, 132750688498773858594815616,

    2416620089911382596489159040, 39759335792103129623770371630,

    597171730934711420929585905102, 8258004906440658976282174443584,

    105903632542755640943173914177408, 1267400803448818897442223197361390,

    14231027231411589412685961570123776, 150638506453507880740087424322494784,

    1509479149806996502727436042726569394,

    14372025127629181915166652870982419950,

    130449578263833743547896513195768488320

                                                                        n
       Theorem Number , 360,  Let , c[9, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[9, 6](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

576, 3819762, 2887788160, 762539176512, 104156937949806, 8940108137781326,

    540258741362161362, 24737280785184167730, 902879709463385186432,

    27254195327868188652864, 699551013877639218820782,

    15600807148226046594273344, 307508207685695994527907840,

    5432265425787303509101109070, 86994719577472956602771206720,

    1275117569059564756784556011598, 17245341200618513135278033143762,

    216704225446861048702747574033280, 2545280105736609071907253916275008,

    28089150926334324472040881550856498, 292590653976292193716309020903159662,

    2888368926417107142202821418089578496,

    27118828993293501121763213274118471680,

    242943056698974085388799958078325927694

                                                                        n
       Theorem Number , 361,  Let , c[9, 6](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 6
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[9, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2574, 10783954, 6744846606, 1593358302528, 201773100786560, 16377731535104064,

    947615984532328320, 41892593608054299538, 1485125286763594036032,

    43736334196877960860800, 1098966398392045429615346,

    24056601826839262464781806, 466449431902315605941944320,

    8120040050269367736867081298, 128333066516225767568627795520,

    1858671980214678524271870024750, 24865071544803998847996253333938,

    309346455852078922228915712632896, 3600107304929361320742160977473010,

    39393071702930906560594401684007680, 407105125078370196949020454192361856,

    3989304137621068904934176394310093422,

    37198190943879272079808148893162270016,

    331091562745405286839403033828012168256

                                                                        n
       Theorem Number , 362,  Let , c[9, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/11 c[9, 7](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1385811, 55066229920, 228001582903680, 280902675987044640,

    156941188593952506240, 49596928146779954552160, 10130676177135753418413632,

    1460480629822774020232296000, 158061933749996479952641772992,

    13437074120883289534923528395520, 928711363123272220400087923318080,

    53609397425966421739966871270586448, 2640742814506852428140571908252656960,

    112965145264550063364829624705115638784,

    4257864036369322510499288789111818948480,

    143136025917125901002891203269198072212000,

    4336085085469686793430499181174377520128576,

    119421136741248204371729050586843393116923840,

    3013173418853095199678581073880098146576129600,

    70117528604804555368602656502381925783608003840,

    1513687586848765988995103247404131382004178524480,

    30472328462838590526518690747158474291421068579776,

    574695936338283705374471621889953712184086816938333,

    10195884463191504221331244768195717392191282137766528

                                                                        n
       Theorem Number , 363,  Let , c[9, 7](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 7
                                 |  |    (1 - q )
                                i = 1



             Then , 1/19 c[9, 7](19 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2454400, 32347076548181, 5557727444940428320, 127369540456823851379520,

    845541417265816538029224000, 2367002901912800359038759012928,

    3469177520016086530400114969848800, 3057084412549637066011977567367170880,

    1780992304118670166703471417545341176864,

    734582425312721006223308327075058392342720,

    225817838542438518184800999706105019796798432,

    53835465187810213848906225835423805067708256640,

    10271667375019828932681614863952259414012367812960,

    1608891794601901444855526860980447189757046030706304,

    211257844265971218632001886156863082114006969625860480,

    23663494664624522151619508714385288682472834782635934464,

    2294705591749983041191518737215157765810929200261126767680,

    195084655485057360489726180717035718247552398322290309496320,

    14698692535716749246354089897049401022522144527937295627454944,

    990807118473000361949065535985486720036082669853730068049334080,

    60247783054988359772170496090903915752975611845304071008945642544,

    3328871936781263110802037735106216100310973538099800328977680265280,

    168214219779904841130433762845372031368826707019052172244455825629120,

    7818808250264169619658653159014310474908427490001183112789903086853760

                                                                        n
       Theorem Number , 364,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



               Then , 1/7 c[9, 8](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

23, 1060664, 2103993056, 1113483352792, 267367325600408, 37271224566982600,

    3466457443418541608, 234855977218367070599, 12305243953934981043544,

    520467849178545682622120, 18350272248283514702665216,

    552809898685653503253584744, 14510587760590691224297084352,

    337150427619519770539477802080, 7024579056014804585958089183039,

    132667733043304559543023799173560, 2292014668041830087582288794806272,

    36505175880484102829626438756672480, 539618330762948311750153622383121112,

    7446341087710872909388452685054156040,

    96413364516731491477455981923812104904,

    1176579994815972792310984584490523788494,

    13587101480670911703961664029190908619200,

    149003839803092862802034782021710618231296

                                                                        n
       Theorem Number , 365,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 8](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

102, 181657282, 3627057271562, 12039735695043028, 13912128782974327290,

    7830609788867715209528, 2589242310008918792110350,

    565585707855846449787080192, 88361082530522842353695500288,

    10449736851997331477855921771656, 975841507209861077172252574125938,

    74333775040376055503613118708448960, 4738580692179646396883560799579917480,

    258061506788718596529832098461605525164,

    12211059403715558921510341822205905392174,

    509151843867104278620178883409909063697728,

    18930107170491293593057214353229725515683042,

    633953537163860948628109588673642623107382122,

    19290138721884553176782803105317214354960507506,

    537357664585020394197008399614368383305959875016,

    13794556227641752274502161882798471565132516213404,

    328240695231308481262292438468592502323372597872384,

    7277036648415457247442102889622840498024954103886382,

    151003805869628938911654552576986359654678924647927028

                                                                        n
       Theorem Number , 366,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 8](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2873, 1338904672, 17550343577152, 46343496712054304, 46104915374207552064,

    23294884697069919448960, 7091165086647287522424992,

    1449987977311179353658992704, 214550272121512581252394964960,

    24238477575158196069230944359264, 2176437832436013188858044155433248,

    160226958964453740432417007703310025, 9911761648169391338684058769710503936,

    525558530076938697723239273062627919872,

    24279499710215252790660781906433557560928,

    990656333933300293485609936065655178447264,

    36113314964457377794660481477312280793865248,

    1187786621717943072443318350638036413070542048,

    35547823463405897187284410755080134941949352544,

    975185360863438056104118738810123577697957774048,

    24680850726746606762468861431448924350044351894400,

    579562349850353883954771862228391358531402941409568,

    12691036893238992802851441196198751306597334772649689,

    260318177179629008848167933421514574277035059735244128

                                                                        n
       Theorem Number , 367,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 8](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12684, 3493744338, 37867749255002, 89841251881320482, 83232380916995494878,

    39927228393980386412996, 11677445976180418447150592,

    2312355410791438663989672358, 333202788616134123526206491328,

    36810114954887190358267085888718, 3242342164704878221732411178536308,

    234731776065970585084707460954735110,

    14307733779349797383534560767160562146,

    748732723973800867834262917403060592678,

    34183237837452764655714284271590671308800,

    1379917729553167105575080361018171086708356,

    49816109214220717306669546522180418141037568,

    1623939731703208206200823370882373053683355136,

    48203938595526550430092586247753541944517000256,

    1312394749649193736019824609886602100002428651456,

    32982413279302443369624175965851535103224658261528,

    769444577114647941440907180908683508857353615302038,

    16746187590410102367839844800700982978210025043755822,

    341532568169628331131215770569964620137841671461128922

                                                                        n
       Theorem Number , 368,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 8](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

191826, 22197653314, 170142843563896, 330111896037935620, 266965587621047676402,

    115890312360295190379008, 31366345257720736199202346,

    5835103520647484265763316672, 798421501712045358054484275988,

    84424921027557446981924235837720, 7161207272921507971728049734335038,

    501652544873905583966564862824797696,

    29701417017722139536805199406542826810,

    1514522805855948385329046367175885311784,

    67552402615057440541663768879189727631530,

    2670024919746723773552657802873207435753706,

    94554025716062685547433884536650043131021224,

    3028499875973767315712148427799121139408234104,

    88449223672062123126645388034512296573591025664,

    2372246350142527127701225794606958001363292507890,

    58793310087636621659082814544911281860187271868608,

    1353898240932955926957899274950957134038073522250566,

    29110903504604472277018498075227735871252254274487808,

    586992077991032973543505462089768014908734886536462016

                                                                        n
       Theorem Number , 369,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



              Then , 1/11 c[9, 8](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

674968, 54225638464, 354688623066940, 626011307272090166, 474458712863694279680,

    196288644701253741232564, 51168528277368583876494426,

    9234010393103167142734508224, 1231982444242445068430146905856,

    127506994364803571896359921830886, 10617399324967410058109415767469510,

    731832351228612332288939461408927210,

    42714442312356270580128973741062613276,

    2150445535793021889516993887173837158912,

    94820625329240912692203952195634029783552,

    3708975509841492458183484888765292173526210,

    130104871836020443430389596478619868620810422,

    4131033676791446042501059348931818999456981908,

    119685182816715729093263639458586396252537273926,

    3186263043165273550454535637393149449161226711126,

    78424994770548198536401586123014830930468639541248,

    1794405376433086963337736430250913930109626112762472,

    38351229394997815734828441973541834554987606277483902,

    768966061051013699398479620519073619684134506170398720

                                                                        n
       Theorem Number , 370,  Let , c[9, 8](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 8
                                 |  |    (1 - q )
                                i = 1



             Then , 1/11 c[9, 8](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2249630, 129969647396, 731713365806290, 1179035069799095230,

    839063527921930242818, 331206813113619979850440, 83220436234903244859549382,

    14576495979483947556220820318, 1897006116680814951456212185536,

    192229105415047167597984943971946, 15717131919764890089726467672210432,

    1066165147978629851637685828196119808,

    61353959237920040031108352133334975848,

    3050042855018541824589763005492386355698,

    132964424359887670407228401712444965888856,

    5147567723834071241332615150432020658917894,

    178875073427315099961595237974320816905885696,

    5630687502171017296923400393858760288660895762,

    161839065414745548911479190335639524444658333146,

    4276831298880996253761475812632183667389988864248,

    104548841169948711851303069304322338129513136392194,

    2376898095115461417530177646041963668141177021590016,

    50497961333688083259090526775606954349801617225146368,

    1006855078629925394180227940073037471728330161202656384

                                                                        n
       Theorem Number , 371,  Let , c[9, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[9, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

36, 158400, 81373176, 15370430256, 1561442938344, 102649336442272,

    4861027735382340, 177648927693483672, 5253867627421140816,

    130142873408134548096, 2770994948030364931560, 51742954031750181822192,

    860995672882964792036640, 12933196688831573712349080,

    177255223423854644403038444, 2236423091034237725673805536,

    26172731377497692429502634512, 285949096841747500661596974912,

    2932882251576512159087617508928, 28377571398940813045011024779568,

    260123755204136773301710233094608, 2267473082157644258111050216773000,

    18858727635507919816537776406150464, 150102040331283373958249012885164792

                                                                        n
       Theorem Number , 372,  Let , c[9, 9](n), be the coefficient of, q

                            in the power series of



                               infinity
                               --------'       i 9
                              '  |  |    (1 + q )
                                 |  |    ---------
                                 |  |          i 9
                                 |  |    (1 - q )
                                i = 1



               Then , 1/5 c[9, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

264, 617796, 247275864, 40393408652, 3718706807232, 227349022195008,

    10174992781748592, 355267341119426040, 10116292429509056856,

    242667471858227087352, 5025678838024299082056, 91598859945596396955816,

    1491889759698339977263040, 21985470334585170705339216,

    296177785629043751790391152, 3679000582211427552575465616,

    42446599843609525994928961992, 457733832412049740664092878120,

    4638677263237463449129980721584, 44385294865648895992606561105284,

    402671593023579387235698936875712, 3476354015868143424080898469602912,

    28653383165077568390880353975053920, 226138438954049233879243480990913208

                                                                        n
      Theorem Number , 373,  Let , c[9, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 10](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7429, 95954717, 151675800902, 80711625688372, 21288230116132101,

    3395866843335733194, 368983378865904389646, 29522998217994373151331,

    1836812224397241999112227, 92477474567447883094064287,

    3883401899208172804944732552, 139266986134373189390741202975,

    4346430341691063541490466725671, 119875839975098035113824867190568,

    2959090353803818834485433983897056, 66074715570206139863738664255712320,

    1346744988786380036379403605044026952,

    25250569315454052463664337870344257820,

    438431727386266468312704893467647358854,

    7091124319071387296102060293467223561699,

    107383990495399501856694529763616552055719,

    1529488322034214656137710352101070470769239,

    20572538408945205116041712935938747929553083,

    262260002843248800279298537608976190308697981

                                                                        n
      Theorem Number , 374,  Let , c[9, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[9, 11](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

356, 1248361, 701063972, 154473698598, 18666217504700, 1467552749749091,

    83111877090625940, 3623989500561578418, 127454425377555788764,

    3740596650505680214915, 94005215135999217678028, 2064120087736033884301813,

    40241657839134996188495348, 705767658542808502388381398,

    11256344055757683053642915560, 164751167000200100043540018601,

    2229995305224687529922944573564, 28099288383141805087141333166347,

    331503095551713616570599477815788, 3680006442301171375188139369785485,

    38608520525041343788551136417103864, 384304993350304243613365646561269055,

    3641882426067837628005923222478775716,

    32958942630555775799869500929661529340

                                                                        n
      Theorem Number , 375,  Let , c[9, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 13](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

38, 7083010, 42413931819, 58191902969052, 32809485898927532,

    9997401644976678368, 1923480170010928733792, 257915693730175856302876,

    25783661999209927279093166, 2017313899983509942130986478,

    128102581467528108820044165879, 6790462700622748843619699225648,

    307228827831422772455591695881326, 12079751973608264643701868758749946,

    418901599718776548447955673726538838,

    12971331035793727186290491260722018972,

    362413817674982830197915665059960610374,

    9218095219545587331912724845277233964594,

    215094788353206030887412984252744753014759,

    4635238075535349298685023718237227303259700,

    92791843874829369658647042599674149600407464,

    1734545861172634898763529040610078109288724780,

    30415186148967675984571684528311895792788913421,

    502343018321127943063563521238377820529168210452

                                                                        n
      Theorem Number , 376,  Let , c[9, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 13](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2361, 105043020, 374909833476, 383429124350646, 177565945208369310,

    46847077470654487912, 8061926414868998679646, 988180745058891272996630,

    91709286861996727501208316, 6737738266912362311253439196,

    405292005353330330855025204694, 20491030587297967927632619040364,

    889156392441423744565413333401038, 33681147154112296363556665139000268,

    1129489312314254303609491455790942752,

    33928632125001707208345832819996463846,

    922071696326384877626372611951467634485,

    22865568299420862250297485185014502160696,

    521220130867612438353593093216729951488960,

    10991929129399706136272022244758136111878690,

    215670775930546215414764398193556302071580827,

    3956758307940122162029842804810111223383093916,

    68178020095096006481657448295947130653645413800,

    1107712867366897626644020343059998677557126639188

                                                                        n
      Theorem Number , 377,  Let , c[9, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

14322, 376207351, 1073319722784, 960682338778358, 406016624651002790,

    100087424389505997286, 16333717334650576372656, 1917791757623620197142556,

    171722876852945635959282476, 12238126101533628318946607521,

    717057461008858832257873358580, 35428774814773342445836062337332,

    1506341005920154232748616669605380, 56030665045894284393194989674656085,

    1848411295597143590509484515947292904,

    54704332503712767543478591888860361400,

    1466638467803193533100096635927146705790,

    35919419038902746051006937021247201702914,

    809430492065927143564732978495981852962836,

    16889350249802181462773211393021656709323516,

    328123035156427541679256978658434159709708052,

    5964585265852372251001977702221628252814686625,

    101891533123131054145033609073551791082724295924,

    1642119971136856382769640368201700831794526102390

                                                                        n
      Theorem Number , 378,  Let , c[9, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[9, 13](19 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

15626185407, 757319503958435560, 706554858071860072631468,

    84098912012187694225952374156, 2655090851769366206023786000478087,

    32429972028823549399750005455537179344,

    192028469267015108867113244869321561074880,

    639043212010970752176037014961607724474793340,

    1325117607899418691507689633556841556860346401062,

    1846350355239238494365913208346553018037624082610792,

    1830426683974039111052427043227643353835114324529779524,

    1350018730184430266899614156514195952865377670995209630680,

    767594100124677269065271210160156815907759953231158253509033,

    346334936507518058500287127807637442241163470481091920769628944,

    127004871236769359687719398851070960444389939529413824224335391316,

    38619443548005252204339597023854572800400153734618174184785360223706,

    9904469897413574624138234476861591689273839345490024658349262598804812,

    2173785409684882867510759259444250456378742392461930525400201132493573128,

    413453600208147374870192931435842314430761896889323954191878622187833861408

    , 68900811609478402372421112090457220605019797147529149080284204577571871\

    142104, 10157591798690270298260287940028457661252824752273267419120217094\

    951748511789515, 13360449161436744014592217758017475294650957732869029898\

    28360946822737313431449728, 157979152546197464115701930564372437973571822\

    380384952917286470183959809968973077144, 16906832443406476102420338629449\
    959460813468072587381292283367179976068149495736852504

                                                                        n
      Theorem Number , 379,  Let , c[9, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/31 c[9, 13](31 n + 26), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

216928270046726, 3064267852808222895018456, 200777249906127942321222365606686,

    833263706820144445205802511334259655848,

    591942459895493753728367089945798274768548206,

    119636764294843045765520362754147074860205283023944,

    9303343258066625702026548084940556340313037105777321551,

    338951723277569101823192465201577261956392201019846356414936,

    6635589301616989765291686091837720519599483981952782417960152554,

    77140262867921955558835591396457302403890482055850610190084835653756,

    574355666756373571781440321078186410870563956599135580509633432578818832,
    290494139546042936021521024251497836416852613194876326268631086688967841\

    8648, 1045950387845228626428324974587467338951412586834884130902727376338\

    4748349549852, 2785078220756911365383760226908450613010560897213583215718\

    1360098737632702110495336, 5659328400215327423160092380095498237370305686\

    0710228273836133541338602503397230387103, 9009832062496458621091976732432\

    0977480212195406852440797348385631520557540875797508819288, 1149128514955\
    317883378203910601786268976553298138540521853020595023045384064523275799\

    61699554, 119674769542382979929496739057286690613865387373890287772028917\

    236487813276328814107448564903588, 10346019367911784799413454767551657934\

    8929438747138489957254239815996330286385723771242503481302121, 7532031132\
    414387410105239576019651115079724591238623714257149050556294850501864965\

    8643554581509888240, 4676126867018081600431398144587012647621464067425457\

    0525915838376466725429017960491636332777198039454212, 2503365078036677218\
    781810397822301378340664618630477240294848702550122199798828560976882113\

    3874910526949692, 1167119662743627491389739263360300371995855359015098328\

    5555376595879213300220585325068757884417832127820772419, 4780698890330709\
    113654591578891946287893736752363673286998987186403187161680071624117075\
    388453839443124609731280

                                                                        n
      Theorem Number , 380,  Let , c[9, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[9, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

58, 686182, 798969328, 309802572748, 60392735895180, 7246372621734261,

    602206173895299071, 37407542632004274616, 1830879922033615598497,

    73365435321902687834169, 2477516346312812552229477,

    72110890312467806867954727, 1841703042808371916255672917,

    41877994786320460476035318104, 858046848749614620515065943420,

    16001389694712422977926743235708, 273919425515016468857622813684078,

    4335829957217712713608103449863373, 63861973314077966343205828882506047,

    880071289926565779259693339020682505,

    11402298027430060640305895628127742606,

    139481327177924491524844575804143374311,

    1617087984285825814681809221339932485431,

    17828516485595035414897523684193041983357

                                                                        n
      Theorem Number , 381,  Let , c[9, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[9, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

529, 3184953, 2822055196, 931892045195, 162806696965347, 18011666589935733,

    1404926849314338891, 82903919086169410055, 3887988411722546913481,

    150243059577164646797265, 4917033429562174692537836,

    139240462653605295091477819, 3470789618637961283563560496,

    77225316450707646106345633900, 1551614034497847647187847644426,

    28426083456671593852316798464321, 478785264319596994159807192358908,

    7466682029254669379545823315486186, 108477121225999206017837409043544173,

    1476027317796223946196738453523300849,

    18898956143822975829781451516569434403,

    228651834911745203712264299075083222666,

    2623690340207427654228835775038349580066,

    28647701572642253552648347789741093627386

                                                                        n
      Theorem Number , 382,  Let , c[9, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 15](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

45, 13503984, 118468802934, 226383181904304, 171949159526592659,

    68891677633774074072, 17103635862245847192420, 2914901376216154238229600,

    365744954483695352112528978, 35535612289148790173223890400,

    2776556057275029367125907233139, 179641047638251283082922453210128,

    9849757822263506302050379985685387, 466344289082692503832954476796645824,

    19361822909642310664207225074304799091,

    714060295456795184748623609411413511472,

    23648188751702257191854941151460837198651,

    709860021541073141579042989809245481328608,

    19468934158366027133780476748542207321722453,

    491287800566805491068747045422621847408149656,

    11476503490725686945114581135259894490315312489,

    249520474642911983559817466373970922078568285408,

    5073480286581751709425864749811121108812623193318,

    96887094379382727058135343334927636338493004022976

                                                                        n
      Theorem Number , 383,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[9, 16](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

670, 5584007, 6521561335, 2753938752388, 602305548720595, 82081791276910118,

    7786183860103064797, 552899114252600961380, 30927075313931267145313,

    1414619464972401912937360, 54436792428667465905655523,

    1801982695868999913408379908, 52232000007135229972611005120,

    1345076301104600153212530976949, 31145621389027085532916015181870,

    655037808967363194114201835744716, 12620479192658739154259806388128020,

    224398117783002754789732459919858635, 3705635554117108972519662097200713162,

    57150697198958148944139413244606749485,

    827217191644114322958743067250515581229,

    11285907142883698994824334026637752231874,

    145695330653056914177595062260992801827810,

    1785848224407671275368840903250200446120062

                                                                        n
      Theorem Number , 384,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[9, 16](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11644085, 11541270532177, 595204231745930483, 6394739123501219900597,

    24743996558485211775297965, 45878099740643099763999157661,

    48407674956197566316763917926044, 32554983629819121647526924767287210,

    15109014263792514545380358720931232106,

    5129957792219863179465606375744432832670,

    1332052474409024065490552338900599815907402,

    273858187915029871954767192189188048295066246,

    45832853674427407820418837237655744876637666832,

    6386965761814270986796107046321389831425569761641,

    755107752313400218447838308974670793011562680333275,

    76938366055972938077658764049823424845757108503040898,

    6846841302461206837747968673740508049839416031997638721,

    538295709368386380861765928464784153036633286429830791201,

    37760454012022635348109549746835973656636794735612473675567,

    2383928923578062331260951687314290097145359757100097049302710,

    136486173561538450091692023991819105499716356648589012475828907,

    7134143436371623334375280043825925333800156458828353553429124048,

    342491323629567553043277192677930512287966514763525757949072479309,

    15182194029543753341461968243459468318511287536818223875802466200737

                                                                        n
      Theorem Number , 385,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1556, 21905784562250, 162617386544882635700, 66029018677496461713475266,

    4925612202229203056947537654149, 116304109644214781786954190446308061,

    1175063097653265164533587632287964213833,

    6131982894493829417310952768703342299923750,

    18766840023086548409779976376628942299505553727,

    36880433104139742418740897776972812368572948932421,

    49783648624453077579661553208709274900195776090202934,

    48611903358128124451983803265871504165785202742889631869,

    35766705594474400368405446641523603800099241883424970289445,

    20490627776741295391202058535729993185305272913499680994619648,

    9389759417816727897624076404550584656827024233832849879031525869,

    3519596961772012841510595014458662260412594403645453194061248850477,

    1099654771601973445860696507197837457253441586017542732705875183737734,

    291017072962852603373199856034089925018857714518967791055602114414855265,

    66142254185907478438951464093104619133782806589452694500230190484861200436,
    130660608028428638730168263714813792562959871970551311267499402499630876\

    56898, 226709168579061250148733207118774580830218641901976680628759996312\

    6634268744146, 3487082300380665460744979507617153497287889691987101385930\

    75769120637880432887645, 479377208129599156872652443995860288339827907238\

    74025094552525539590011614533298720, 593309396882685905680061405452663387\
    5552039139176955161584500753514260647078618083051

                                                                        n
      Theorem Number , 386,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

62114, 177148690800175, 834958370843813215584, 266156946871517242029836140,

    16970975626703981958167564081167, 358044246625004640406674706522069712,

    3319384453789323233771862949011103597610,

    16175051589136026795381715324813467607912900,

    46794413797504523375652331541025362775691820568,

    87709115515814147182867163600495018719821043369951,

    113692154047506372835297486064790694449103934575092348,

    107172280049954849935806542641279949681485362581489617945,

    76445460176081334326719019810343466214663827560346155821064,

    42604487237596055529645632048948334899013852722546321489139336,

    19046592591295723765370969419261953626017298128173077261903531952,

    6981583277524636541805149045802915237215807015660939493942746403935,

    2137438503890224297051342006160919356707426612427268045320162515966810,

    555244760933414377422141734803696024653935479369514322134393120724625764,

    124057897626169391242406091130040918422983889019864650459396743344119490162

    , 24123303433006965673200414817465364307728192837350824275638680829883360\

    169894, 41248293090583691435649744044756290589421283643984237107284505908\

    66462361679098, 625870009297978549502393998686781465345778631816701722661\

    926040065164760284277550, 84952425798362245680177585200303217478968316257\

    246625345829005787000230352670695080, 10389770244541643613224180996842878\
    601199343660512141793318404730151046503884861695821

                                                                        n
      Theorem Number , 387,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

333165, 489549057207245, 1866703007180062017843, 529994910549705856884817620,

    31322613206951366440259005716852, 625550847273483044511714367170370071,

    5560447301793643563064822587690530634270,

    26199588367769051530256919630734936767455629,

    73725512777985711192013520687238276453667766040,

    135001318993077507593089024001404899406257309626435,

    171527737369038175858936960450478752738316218085007001,

    158899956950783038672547476964731572986163860460972931963,

    111617332021464194278191773124121084123532623900995874821087,

    61363229333187420515863830791173809533877344991368001256917580,

    27098853466744169338097914337181769498811548654772323653520473497,

    9823772607351840397416567686944361853154362523823556625519242835771,

    2977430950022335493155581611640968499155636216903567565107510840252934,

    766350653138668312470132720914297734712029827228815105296566544053121388,

    169778931707374185168632548413968199954369247712065416721473271682810327400

    , 32756145255789167060310650870803084975903348182475091929887190827499953\

    722615, 55603856352088017504232318073899704781652536366631983565099138256\

    06547696232502, 837999574464155501055548367401417166463834880144410249971\

    404088662284513517365233, 11302900452996260793861740001311251799942772656\

    8952506361346569124321056306786779466, 1374193112578942376006175871337260\
    2765269531470914154246836864343728012683525287143475

                                                                        n
      Theorem Number , 388,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1642355, 1329164544590673, 4137772374030201112151, 1049791884162475401241590536,

    57597311331693519944026625063546, 1089892810034443815446959440353150930,

    9294256274643247633207214871568416553743,

    42361631930547554418505826703364540096269055,

    115984118555305104491422497130472041921666007596,

    207531421805066721766755999408061866083265465698168,

    258502486763484752328238343265039583118451369817321590,

    235369487844345420173837386017300581735155047175632355362,

    162833589595996271395500917302105489348304488495136856406197,

    88314582892760173588741897877059421205698819007910537484359881,

    38529055739626938593581025226015703373304973952877834805591117745,

    13814460599289504934831912905430775914682103471919862715487917926025,

    4145189973536252033525722784878671985666452948673892567857873673997916,

    1057171510910036994261270940303104138441709736771183770322204004887333249,

    232239228748343070925958273400240747561484095311051599833052272043535060297

    , 44458680688327305976496690751765036480143021725219568330477176763413914\

    419093, 74924727211793957750644078868396675247861062640452463619111324007\

    25496323699875, 112159691315003997249514818355323095640877606669328687298\

    3960346180752487429109671, 1503308448223510163244884627404783526872675126\

    04800797999133717679662870060230526721, 181695129752995794250611077892579\
    46088345692705312913572498332980314499080144627596153

                                                                        n
      Theorem Number , 389,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

132547320, 24141703316738270, 42924418050620695946061,

    7912517401097404372371776787, 350497607986140855984563129178164,

    5671518997394098366425333293919122016,

    42851567839134386522822077135586118184650,

    177202356886195799500143477298886384190925218,

    447657239528301082624764559871052458510155841366,

    748316301364769397316280867451484449540383930997828,

    879129395468077651338707101823450035866698381682951485,

    760624301370125203393590659967055247859371309640467209825,

    503037621709762857977610339774040274259984297326296894832192,

    262092017689757468622931168830903820820093342242711240391055823,

    110291232489596852243982038081479743285362103032203514485381208390,

    38273931012363471460186486534609015546339271496585616676706443953116,

    11147906878597941161824637858909470572575575244933458687745071476815645,

    2766682393327531832758432743403240746311034933032958505082696226604680412,

    592727294685142806556818778737955633137682360237149985473815228161321262913

    , 11086717325023076779020929951555334328650536420194590838291775648336125\

    1670195, 1828611085297227135558091645147884434517055952122409212344528210\

    2205106965102864, 2683011640054571679072596221310665970626245327323445674\

    801972122979215732145529320, 35293365767193593664553786635119971225792903\

    5081464536497486049585286019892502249847, 4191382971235281396654179078238\
    8785365286074546352675194512889596381816220887361395857

                                                                        n
      Theorem Number , 390,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

515313447, 61600282565007581, 92186398668512631394103,

    15362352943275067639003236800, 635461095131011705359134187649003,

    9776420994218685882304937995896679598,

    71023520492106437875021008717115997289495,

    284544530331606871100067054528545588718580565,

    700191544981753571496485201674216976207423323275,

    1144693865313047810804103871234004300677025380961256,

    1319249343113941439766034717054097767020671222162757917,

    1122450330272108875953789421656812304323041427184867036476,

    731420679257910596209813990849893707650449582254250081379821,

    376083193530929593526740423896163700135487977897157227195025875,

    156390721145151780539801158134635357691236892748323213648340046017,

    53690614368486512637767692290431480146427699496318787422447798412352,

    15485575466636516678140425154638747463282258556992555995994034684772869,

    3808794709189300406791001788693455887376631633763729424591528737572783309,

    809255871381969495044080857052945816390494073438209258625515525204229669625

    , 15021255553140297060220323247174653089704777136981892724892440013265563\

    6044774, 2459995606232686137093203933694895722295282084383657986753541634\

    8783756992944416, 3585548246998579070403638262970400166243918611570452669\

    721991973610419154828005692, 46874159314358153417700025320910554233863817\

    7251880334018802148298680808115759245866, 5534430759934292400453739044642\
    8322154469124116425393001734902944954761972427846428895

                                                                        n
      Theorem Number , 391,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1918106275, 155039101428241821, 196541685598415052265494,

    29685829243945535141411219663, 1148220773927201768281288162577231,

    16809028587928867336511592130766691025,

    117475948201642595412431709342222895924678,

    456144094681410136928145923660629568970562574,

    1093643887235269291624312967435958783843070609334,

    1748919151279684035564459648791876234265355115380005,

    1977634717516266239491815169103216707548509283063788260,

    1654870773896642266661767131611940425222344266105230326754,

    1062621618627147277572412259589523279908122216445431795576027,

    539256710680486158305034105542374993984868316861213560807675170,

    221612032936779004910355830556307978539272450729340196319988074847,

    75271781454889799249517668672212558609961975857622893495635180303320,

    21499198985378560357542086830310449661391593238360293868830476914054617,

    5240781977988845910985851181860386335164661625976175215830043753652925944,
    110436828084654848191640627793271077764235044658234872976755936243425462\

    1087, 2034330120233609927877075111212327456928411907222821780646939973362\

    04249265402, 330805197987363461734454649941958731420298849991085991539371\

    45635825377606784735, 478988642232694738897187468934005732215112004685857\

    5450498375212181352490829399563, 6223302624002517614412048663688372301474\

    44989191567082633404467706498160116993120218, 730541527684736075369047055\
    10448982388783562336935311800372584641567745085395312235332

                                                                        n
      Theorem Number , 392,  Let , c[9, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[9, 16](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

23702359520, 944789234658060415, 874728487320649056488225,

    109333740868437029307510277509, 3711905644899629163017590114155300,

    49314976676033482273460430207183903246,

    319462607709276820407896802372119428257137,

    1166407357138706916043281239078962663811335982,

    2656935042962202914403524267325241996591858629110,

    4067965067910543529021140022758222975834781159267585,

    4430309078063133836189862083008564032249033903057295643,

    3587315205403391798182504806482689546470010534051420452300,

    2237406161137439499672262921376026919270354181919054362727080,

    1106295564976445872475417018828971837380566273421087319744011706,

    444121455849934416687174130968712395295839865026763720755865126285,

    147679777655619591055301592904181971360376834709575505166560291568160,

    41371354221519938238606247807176259502944081377658668650073793840315395,

    9907365198951339859987728731745209513872871913008990499103201938735214999,
    205383307955709032795741878774528930574208225829071871141669817765219244\

    4871, 3726412017946974149879452887505987406557900218714753831659553065516\

    28417755475, 597483924121406591671471397426944533778986072242368969165555\

    54310365467663230793, 853840923642478334406491913187480147933167971209232\

    7687761256928983503357370248296, 1095819739013145239210661077087450088203\

    401444834215085794265139385555529115586550070, 12716274328698464507538038\
    4988321903217333641837314282901390091770222599468363692899551

                                                                        n
      Theorem Number , 393,  Let , c[9, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[9, 17](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30316, 1813765100, 10321997546176, 17017011469712416, 12533733460443946090,

    5165234232719775110185, 1363654881793855436512000,

    252152579486807273331663872, 34765115785074377559394254776,

    3742025372996452585332299100068, 325641471622687255923250917257472,

    23546377224072807357447647697335568, 1445999628977738911013466782615598528,

    76775971368032135598624788359076234723,

    3576998119032203659910815652839755332224,

    148061472881496088713887206023932367889920,

    5502725965446235093324217356512886645275300,

    185293188328535349973404434024028828032125460,

    5697618058377528448657053069800048570684937120,

    161084432305332448441778451654368076609540629056,

    4212648620575337439994301007946214587742492708496,

    102450142188231056596184599657474761140635687778350,

    2328013570953489265935931453960007663799590644800512,

    49638353546624118880156778604188816313540699359237632

                                                                        n
      Theorem Number , 394,  Let , c[9, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 9
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[9, 17](23 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1582143, 618421798636459008, 76742089409028300579202048,

    342815489400293524673315244201600, 213054175713987947141405225705791873304,

    34319118648460917219584969085495830737186816,

    2031421398816508602955862729792237650508929224160,

    55035390289339941056496744916853180376797939276625016,

    792047523126949934546041557724293482425722699459588458825,

    6736602319575647079214814216649437571474275590328951708405760,

    36664020393967632060039748171822932782103125471300040396490277376,

    135743641384927582361242921887913133981828826700502952656624670232056,

    358799659682867602620166113578392235217733740433403767930975294681186592,

    703948957447805600880100566749806758773199158780379329402420926111943936000

    , 10584017330760381657822027562239389230904847174601447522776393549592769\

    07833568, 125233439573658416258323828061471488605918920271328300754261198\

    1356977004929212678, 1192558787300485259453860327211627033434507523183422\

    111074610029566066091697675519334, 93158611891069060230766074666303324834\

    4946296506542547543277168790515906393226696320512, 6068612659732234601770\

    88146356674600369404441821692343797851608535521949425997116737088000, 334\
    408835978903357090988941938330587525610053730142312697543226391248798555\

    067090728327934448, 15783789722135719978023363580247416766947990094927147\

    3789358189404882451171108695107823915539352, 6451602901784109083374868130\

    6181359767456141664386240916191391455599986934363352667322118368346112, 2\
    306118728157816480659688450531101651338345883565471134928904721200600129\

    1757829464617131191883852704, 7271572980140799455555050462046789070860311\
    168883971401608074850676838948548779301814024458279842509200

                                                                        n
      Theorem Number , 395,  Let , c[10, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/5 c[10, 1](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

238, 50611, 3265870, 112787676, 2601936412, 44967662174, 623549684835,

    7256788362586, 73141098097532, 653279910983512, 5261625527681756,

    38736377907871323, 263511620350092650, 1671038302053877084,

    9950357331382701436, 55976208848160262294, 299036754432099933020,

    1523785363239347849212, 7434682311825893330772, 34848823399468691659580,

    157388026482827336743402, 686650776517978962743284,

    2900538957604165309386328, 11887532946704594671827708

                                                                        n
      Theorem Number , 396,  Let , c[10, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/7 c[10, 1](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

170, 211098, 41175473, 3354045012, 159226506132, 5183420258990, 127127121894280,

    2495046077123898, 40845023474573780, 574682405317128053,

    7107398093844786740, 78621808522435970952, 788704987420328839390,

    7255536720249886106644, 61772105011173564489618, 490464840369984973388060,

    3655314252032886589016403, 25712332597353239296704520,

    171524411282225735984497192, 1089621758597977041730077966,

    6615543966354502407041418440, 38510914654686173147056693558,

    215557174106197315270840443262, 1163055627372476531454068533492

                                                                        n
      Theorem Number , 397,  Let , c[10, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



             Then , 1/11 c[10, 1](11 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1, 307963, 647910642, 283431674925, 51982091596687, 5368998976782646,

    365706985201808761, 18088812416887644884, 692629710563339931460,

    21477099277179753898154, 557494755758086088129513,

    12424275221968476224283032, 242469064273612690001361052,

    4209891614952865168117266280, 65874715966316837309466754596,

    938945026820878619678308830556, 12300685453495977619552072808188,

    149242443639941156826579185426852, 1687970565778082572184799229727760,

    17897845976591921148661053847607562, 178788931044257947120121373852836104,

    1689915929266372174428440436879491543,

    15171799792097081182935716193980183640,

    129818209709418981103500447092786604840

                                                                        n
      Theorem Number , 398,  Let , c[10, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 2](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

16, 12544, 1571010, 86507432, 2889808320, 68268700704, 1244966378854,

    18530279401216, 233853054845424, 2571912569698304, 25161913047418514,

    222481958303755136, 1800373615678281728, 13469521797854591600,

    93947046280717314304, 615139300579470569664, 3803392700570981065408,

    22317776055287066998784, 124820720721251702427422, 667893237805306285418120,

    3430346799434282901888960, 16960428121479324741869568,

    80931672402533138723835270, 373574962173967396229517312

                                                                        n
      Theorem Number , 399,  Let , c[10, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

80, 36203, 3686976, 180483072, 5572907264, 124302980992, 2168426119428,

    31145290975232, 381649254107616, 4094252222012608, 39208167358070016,

    340277267423342960, 2708700723599299072, 19970584428358865536,

    137470288434268073984, 889467286278700222464, 5440272452671605811808,

    31607507009519208653760, 175169979137314127123456, 929424078336439764047872,

    4736339025441669848464640, 23247329771751781953865730,

    110177837682121616593345024, 505334583506876156117475328

                                                                        n
      Theorem Number , 400,  Let , c[10, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

334, 99008, 8412320, 369807728, 10610006838, 224134435840, 3747797055408,

    52018383818240, 619560774453760, 6488235116755224, 60856060794240544,

    518648415411720128, 4062840970471294426, 29528318371141384992,

    200659673503714143152, 1283247420016529239552, 7765657006406648889344,

    44679706784535749389568, 245401200924754709584512,

    1291279618529395936746256, 6529742111926093384557610,

    31820085971216769342581376, 149796214116344660965632544,

    682725326265323543717312000

                                                                        n
      Theorem Number , 401,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

126, 104832, 17585990, 1330986258, 60875514880, 1948579163146, 47591988224256,

    938101926445138, 15515318943633502, 221494226722400714, 2788312227629757696,

    31471515889445586482, 322729384556172637440, 3039268246936148401664,

    26519080256384207938838, 215986975956451273206514,

    1652357531447539206227582, 11937746643268705179425722,

    81827484474557689308063814, 534307132163563502738661504,

    3335329312886628851017851648, 19966570918292683578816818042,

    114946169836214276987049629440, 637957840854006693292458324096

                                                                        n
      Theorem Number , 402,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

602, 318902, 43963008, 2958923520, 124863533802, 3764934268326, 87760779201602,

    1665718320397838, 26696886226024320, 371083607319978550,

    4565062112923920384, 50497674154121560576, 508686594508313251226,

    4714841255561104086054, 40553863836528210884352, 326030657766911444868238,

    2464833801365269596641322, 17615112031288664888228352,

    119539708874795021100862644, 773353570187128654135025950,

    4786142102775371323491549568, 28422566583045744157474616064,

    162402595828434745994491211394, 895017128384809247780270065870

                                                                        n
      Theorem Number , 403,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 4](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 74880, 76352426, 20902500614, 2689238763090, 208065712981580,

    11082370674023930, 441913919185273216, 13956390396847535202,

    363347567505938036590, 8032914567643472697600, 154276411397465195147510,

    2620396039305825656570880, 39931484289894541894418638,

    552395407276520467239304250, 7005046091296935897945437042,

    82104407025867340705035449600, 895680878364102903685568219628,

    9149172804022573578166319532202, 87966136829900339354606256616320,

    799709809073414046926897865111652, 6901973363699825297532853671261790,

    56751801366864200976792867019866624, 445989381973798493875585570251852672

                                                                        n
      Theorem Number , 404,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 4](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

90, 659270, 419284224, 89188238430, 9741048167988, 670072804603670,

    32626256326680402, 1211202061457288832, 36069767252943971840,

    893810545538939320508, 18942200183131577099170, 350666755931862263928960,

    5766581542628286926421290, 85385506339139300786330460,

    1151134543018930553857805778, 14261836370209059699154870030,

    163657564219273813616728969018, 1751110385145973345818710318464,

    17571618988260010457946389245440, 166190867619886636504697976660972,

    1488008750898022931674478160230400, 12661555896811426643304574755262726,

    102740691915482756658128797540116100, 797446027444441602316055786668394910

                                                                        n
      Theorem Number , 405,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

430, 1827034, 950704470, 180520949378, 18278394476672, 1189798800284170,

    55515480727468316, 1991651591164112640, 57659974186378501710,

    1395144850963571556986, 28967045597520150631680, 526758167432685858286086,

    8526961888049075128161230, 124499433968918888088633344,

    1657436811348023218198094742, 20301833273604102969624725760,

    230562054821973793050274129920, 2443642000210592508429769849300,

    24307350046639670504289823420390, 228045984490010890680499068196706,

    2026577076658530260316134947445870, 17124223964359885199717729283244298,

    138048499150226244770649272460648320, 1064961981172844886756009418820192242

                                                                        n
      Theorem Number , 406,  Let , c[10, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 4](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1792, 4873055, 2113516800, 360821629440, 33994277303040, 2098458629374464,

    93960772899632640, 3260758652088514560, 91837811404992970546,

    2170905890668677429760, 44177986502143641697536, 789403516614541930521600,

    12582222879491903491591680, 181189670039696303575672320,

    2382378080633306322155608320, 28855296524128177426882488179,

    324360474337476654457843590912, 3405649320329537278753459914240,

    33584989050560000941871371946240, 312577401138912315879122289024000,

    2757238152551023151224983894946560, 23137589179296313979611349825210880,

    185323692614939466209256688381615050, 1421022287637159763348516671860083200

                                                                        n
      Theorem Number , 407,  Let , c[10, 5](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 5
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 5](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

70, 41669100, 356495017351, 574099275181670, 348354400050974240,

    108952387819116930750, 20901071290830955404410, 2741554051916025922422886,

    264585074107144002382941100, 19796329025472567561591103880,

    1193809734056447186253905083380, 59782512247338447583740340971110,

    2545119761022462561330870053527104, 93874768139896103918685023286766255,

    3046637048425422798325068879049578560,

    88128003576848239691771233971542617160,

    2296867793667553471670210262316472953000,

    54437196409824736041658505838156428473204,

    1182607882942932246185832172629457511066030,

    23711653080286510806900051282736524743962440,

    441441934933226006699068737724918543970953580,

    7671455336073315416117065970541333077899219060,

    125029737035273270158647419068562140507673459188,

    1919096052063283854491340603128659758306487829260

                                                                        n
      Theorem Number , 408,  Let , c[10, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 6](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

995, 868352, 182066176, 17750982656, 1048867880448, 43166236204592,

    1345937398617088, 33612368890462208, 699074813117761024,

    12461281171548504064, 194582027028479551466, 2707485733829604147200,

    34032405827618673262592, 390773170934915974610944,

    4136884125709462015039488, 40691684825928790759331552,

    374353244529309022173605376, 3239329478628449907145736192,

    26494206353553678689250473984, 205694790692384365041604329472,

    1521606031887145886443858427501, 10760431528672391988003920543744,

    72960672242752273441242839641088, 475584247999962688769979760345088

                                                                        n
      Theorem Number , 409,  Let , c[10, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 7](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

32, 106080, 42580512, 6445311616, 534402366048, 29080180203840,

    1152838925704576, 35600276276671104, 896698948711966272,

    19046546144390817984, 349815910533942378528, 5664280252114736282880,

    82115006437652377810752, 1079175066438605841549984,

    12990205723460996100333888, 144452080311056533625508128,

    1494752263051326248560383456, 14482675062227335753073548224,

    132096006625774091634093122272, 1139507562623080061060203100928,

    9334844319460852416008795277600, 72883067771361085662568208288320,

    544086202474663254883464209516928, 3894690832683174570046594040895456

                                                                        n
      Theorem Number , 410,  Let , c[10, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 7](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

221, 392832, 123571776, 16242765696, 1224250905408, 62105949048493,

    2331542072373120, 68903975975392704, 1673445636556858560,

    34464587472702320832, 616371335115946725728, 9751088836931789041152,

    138487355872765042292352, 1786979661907164902180160,

    21158385828434097282561408, 231793700100163092251623853,

    2366089087494478182341441728, 22640565693473561059203390528,

    204141892403977253063045740224, 1742357509818488972751455964992,

    14132979700846021893418926515616, 109333027358630883084410505509184,

    809189487873080654577337014900608, 5745739266820686111672594661915392

                                                                        n
      Theorem Number , 411,  Let , c[10, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 7](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1248, 1365984, 346746432, 40049840352, 2761069946400, 131063685096224,

    4670748940627776, 132324867957461760, 3102577606119489280,

    62013378954944489376, 1080737156491224609888, 16714412950584243139744,

    232666938755725944816768, 2948870256305929749307584,

    34355875630744635985202112, 370896948471389719381188192,

    3735692841290755480909778016, 35309716566187233460039804224,

    314790603574468865910364519584, 2658726259717079593398742035840,

    21356878587984452257390390833632, 163722290447911998955419040439328,

    1201470243486757386827594290994592, 8463358303167479213233111750312000

                                                                        n
      Theorem Number , 412,  Let , c[10, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 8](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1103, 16680064, 24256959424, 11250083664000, 2542901387926528,

    345928572177219328, 32049079251106341824, 2190574179909592747406,

    116750791191425681277760, 5051254919579441269188608,

    182878435187615115114839360, 5672753172608947533821959936,

    153618544193009905927075781056, 3687434293456499620913429150848,

    79449280723054221513570091434813, 1552755444270114176617790924969984,

    27773240129727844215635291270533184, 458106036857175715099406358736389888,

    7014173392174004575459053608131456576,

    100263838328847075069182126760075200512,

    1344778446478900898780402418534205083904,

    16998947102846158372118994844816913279434,

    203315037329668829365185743736715350159936,

    2308988654828514058768458250128582144006656

                                                                        n
      Theorem Number , 413,  Let , c[10, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

304, 834288, 377314160, 68323925472, 6887985775952, 457156641547584,

    22065966900629600, 826608855326454304, 25146264061413546816,

    642129662616190944768, 14113784866124300613280, 272289687395029695089392,

    4683442506453489936775328, 72737813357512008961901920,

    1030802378678145888685028544, 13447205324430623831531323504,

    162693965684596704052543065104, 1837264652493941572423804334896,

    19473130554044522843197924461216, 194652042627913904460353972243744,

    1842815008528974942117456571393712, 16585601588603712921636601857140608,

    142381792826030700808513264169248160, 1169351801990769751121780336370388080

                                                                        n
      Theorem Number , 414,  Let , c[10, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1888, 3133328, 1132054832, 178926236160, 16418427972432, 1015774010247040,

    46400071232464544, 1662053623742814944, 48707389594424106160,

    1204858287221272069216, 25764007687198891971984, 485214642852318313848576,

    8169458395338105798357520, 124478373593959083101783232,

    1733932500064830478277978688, 22269084104414403584496086864,

    265610240630977836939904289040, 2960440383723035589797885145456,

    31000836290387737983654528792320, 306433785241602500449462952375360,

    2871050222773005802360155615214880, 25590249456579369987929763380996560,

    217696897302180291763886296208024672, 1772718693661043223447640250579619376

                                                                        n
      Theorem Number , 415,  Let , c[10, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[10, 9](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

80009, 176407756448, 8486910346396000, 68889370972568730688,

    186621016481660889941760, 234913938662613297803823264,

    166329293899461344882089143968, 74866170241505347869366912401440,

    23300702245670711973289644465736544, 5328467414547111929661994742300958880,

    937124884195829090251136179830854045504,

    131306844461916759049419611357978030814272,

    15074084684680172783081361851679920547592000,

    1450326727467223917230274427137806858657739307,

    119148015348251831534142748668833944124927615200,

    8488887072919471138502281785443158652563772388416,

    531456923066648070956117709487731764618517616851904,

    29567660822135103264770957731179645520801832718577600,

    1476061383388839717858589957648174337052831116532802976,

    66678997866504182275895849579860489370390104262391512800,

    2745848649903809002189541408092832959447857438835077906880,

    103751590851802516917338064810383597342218189890428813614016,

    3617871355711357067273382328825237133973440976668372937327520,

    117028356598610972055289825871259377457938495649848083763376640

                                                                        n
      Theorem Number , 416,  Let , c[10, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[10, 9](19 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

865617257, 12210545061174880, 3714153912137973845600,

    160730589611261730076300128, 2007754385395591464788650493728,

    10371091036395424801649591610322752, 27409707691858379011316531848154530240,

    42575053525026866139823297051688101440640,

    42796787383640136228126773904572045243968256,

    29864731739339425634629704764687698979739877472,

    15255912957454592917043632423262140872075224785760,

    5945279973737353323915143751798893012187960961164480,

    1826550870728457334257395806819611042104714233127956960,

    454361950423489997403718244334048181519836424762768376928,

    93549764541199706216774972480190869944302442418784772963456,

    16238765891083183000380826708803393572821988705099432867884864,

    2413831426741557849618396953977988713588224354920748669650881440,

    311387807352962754345233396336386110506263039173808454369860723200,

    35265029902493708910062593924545174249761515233900427433403597867776,

    3541567074963337611243928103754599267241686625537179057090794336691915,

    318185663026132818784061928400799307234586731903534567073748681100618784,

    25773753929547467870540618435534849980812491176635770119652060822478014720,
    189534154013300584527502941747403702980508251757523346951934354334145286\

    7520, 1273181076530433277670244619728854355492433812736671202767067710708\
    75656378528

                                                                         n
      Theorem Number , 417,  Let , c[10, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 10](11 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20, 287267160, 17052478577120, 130445481902858624, 304972113586585959360,

    319826959167291533556480, 185882574638512440591768360,

    68293939806911355357245144960, 17331394921701826106767914265032,

    3235965451650653486612689156383640, 465853580215193699025991068110436160,

    53602720627870539126593993928084779840,

    5071221508517057827947572225524536412640,

    403568828868620323495632798389275376273648,

    27523315380867999521429866064040445518627520,

    1633782206789738056639482777994683990969886400,

    85519858402842739201184841399588539055041132520,

    3991600442932141447426551921257231740380659758080,

    167720681840651479558970235263493899377555564169500,

    6397170703826091381810269943855753362948430953881120,

    223100511650836020801154725152918144070101616254660800,

    7159742177111736501424807133936666519950040619455563320,

    212635116521987592580039656908008258193162423367889047240,

    5873569958502614530845767020032157130930457019659465470080

                                                                         n
      Theorem Number , 418,  Let , c[10, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 10](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

28640, 21898409720, 524559556134720, 2448368004145133120,

    4141004339580234792540, 3437550765881285727918040,

    1670082152589903303906390900, 531730998860519979637538349600,

    119933656400158936363273428981120, 20275902183285382556322804567131136,

    2680605366466437138431593977839141320,

    286402421673755150933128500264097131520,

    25383211397891801497242063881143232739840,

    1905998303729463908351824646720496339570200,

    123385648121909072022139333881753072718786312,

    6986963037838885768205211058030278717920336160,

    350376401693849012133312435557741639387414966400,

    15724225233666162419111854535791654827755068425080,

    637281110823129752639848319791603341138058112159680,

    23509742906501668776367666725937427546759368064495040,

    794921883183885549152967905933456596848792831335477160,

    24786336825468613645644935512101146371502345873952445440,

    716581666499194755881718740053673293522447102203216933120,

    19301275533600935279900100598289441313264253904597579452800

                                                                         n
      Theorem Number , 419,  Let , c[10, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 10](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

128960, 60383353140, 1193942416605400, 4988737729126385416,

    7831249858002198390080, 6154668591212964180885760,

    2865950662442091129623469880, 881835015882649539491981544960,

    193352972294127877804837925261120, 31914947226351673363148085969738760,

    4133312339931255188155137617801208600,

    433741810831639972741162801090279453780,

    37835939541175067803289248631853981053280,

    2801118363981796423690293252085101307810464,

    179038104245793940289259279136241857472189440,

    10022210752403259676500087943302844061422646560,

    497334661602550446664201581825569622201506607680,

    22105685611735890718882171891169720614946546246440,

    888010068004135916705702319589492223134630293138608,

    32491988988434511747784568363541064018399503955014080,

    1090307908068345472410772710096961630173719451989551520,

    33756616403612092792188958756218228702911943251074514560,

    969471374254165279152435189804275227208766189794693303040,

    25951226150409316252124558785268001462270527056944819580496

                                                                         n
      Theorem Number , 420,  Let , c[10, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 10](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

536660, 162633335840, 2684168422635712, 10084714379129898040,

    14726694044988214175560, 10972312294032646800592320,

    4901420256280668107089640840, 1458385882394205201782427790384,

    310989229023686704339160369347040, 50134903621772477653742544673503840,

    6362244444030623948261174532949198240,

    655879411032741872294671367409143312640,

    56321567384368989388798926697803131806848,

    4111628370039150008563925499097096443822400,

    259508113435077575092633073309080549241250900,

    14361725720160323615498525457777337471815539800,

    705290016371003534846525566472539622223771275200,

    31051022914387536663675350386065787158132713687680,

    1236430889524623227350910031370814830698364351448640,

    44873952788304170940547022590018888860075806545653760,

    1494462834645700878754778893981640905989007022030526720,

    45944754970663010076421443774592006088672365581517092480,

    1310846765688500453742351548813209491205132467703434968720,

    34873280737043332622498973808841872433509252919551175177880

                                                                         n
      Theorem Number , 421,  Let , c[10, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/11 c[10, 10](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7673992, 1106416765440, 13101808371823700, 40285085832198319200,

    51234276656441035144640, 34440055861896446439970288,

    14194092940737538245970156640, 3956328436192499698874408782400,

    799003108642016231215026563842260, 122991412360605569932425957723008600,

    14997241906733336598741149605681085632,

    1492976956824483234642471076925274291360,

    124301391989095144218965798946801462185280,

    8827119548140477846460538438564981085573560,

    543441387605024228316725701435482156227832040,

    29404327861867533124350715982839988823064489280,

    1414599587476134041841148694900167476196480358400,

    61114193034605208834914717802158678615392720294040,

    2391544732194014009127635691024955750865042320802120,

    85409761366841694253310928995818490520176281317849920,

    2802193503639900639640018311314912506918154708023626112,

    84954806959166778871563606704349985292960068196691074040,

    2392404347893651325700641435540659154311417678183389153600,

    62871972714062167589557855906654509580845283448807861161920

                                                                         n
      Theorem Number , 422,  Let , c[10, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 11](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2737, 6616702, 3294614739, 693080380986, 82581034219355, 6510536428788469,

    373337301390107822, 16581931197517253225, 596316724739995051179,

    17940373624268938870565, 462956595294412307741415,

    10449771996061558243347628, 209580162976561320189538726,

    3783035169799701693436934185, 62115291356167210223503994700,

    936073240644154690826695437754, 13046038995429824884741610290108,

    169255933816396748528897059488069, 2055730600188049539367381245696632,

    23490667532916031464800374447837995, 253642999878038566905050350192198129,

    2597907189590109635112317809779692608,

    25327317873111387899471747489957090637,

    235751750917017306316774746952419864145

                                                                         n
      Theorem Number , 423,  Let , c[10, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 11](7 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 1173562, 7257445520, 9221886108598, 4650383163420335, 1250497115168623503,

    211240649440669890463, 24835679878466591539518, 2178015295102760879527582,

    149700116411829514421099090, 8366797948081983945711270551,

    391177018642455461352453708412, 15644736811328355494877190569031,

    544957354680041677876650283195177, 16779048237797165332000267462741425,

    462294834022087059641004365275353488,

    11516399198335708738869055247364342044,

    261695288314768468023824308506143990152,

    5465861666356644934054683978950654274301,

    105626338607598328998446795878336256027865,

    1899528078586018644984151474534371202590696,

    31951394452680069509527827810616188991767775,

    504968731955174157659912665790487260217013723,

    7528651370522366496862125414517043799344037219

                                                                         n
      Theorem Number , 424,  Let , c[10, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 11](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

289, 17947286, 63136535867, 58986453013825, 24306092805724574,

    5647617731645184881, 852769025780717049834, 91651271539471528523635,

    7464122854329684459534020, 481982397397428045096591040,

    25532522688266799174262773803, 1139289347234089545508119553508,

    43727748499533897455381025796163, 1468379000134321099548129461211880,

    43747402827650720078854794396402194, 1169972318252087628135010183408224932,

    28366258357979543930261996478918230945,

    628786347978027645425350543605611810352,

    12836528242233000999434450693747697142970,

    242880950651024159607524529041436966397392,

    4283107936521133510895707782184513182182023,

    70742261665074562720171553319851024713263487,

    1099129626283279615493965276849284201722002451,

    16127299600246206211423410065185434447072815385

                                                                         n
      Theorem Number , 425,  Let , c[10, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1955, 64733416, 178930305567, 145500294452466, 54600723430283579,

    11844236569655180875, 1695586793410779091402, 174568695887791924166161,

    13719532038448938891535669, 859592142523672650389150115,

    44368065254405150159645710500, 1935325735602140044508840878389,

    72806163081934920673345317433067, 2401482864510917757614804397441913,

    70405147138825967498293943128450925, 1855647992564364025080227006985494720,

    44396565820708867792744594074408988712,

    972207129252085952169704500629877157867,

    19625871453264811021597872300574903748408,

    367506742443770696507219382385227824007368,

    6418632796893787481556951918065302533189355,

    105065136395438786689195392724685423976668514,

    1618744384189116034112405382277270399265313518,

    23565014543476131202137001621089330446424220973

                                                                         n
      Theorem Number , 426,  Let , c[10, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 11](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11333, 223075003, 495057414990, 353394679553276, 121306757285161629,

    24631839979529156566, 3348922308527444462502, 330683282353151648386725,

    25101445216820209840568083, 1527006892328813005486226945,

    76835022708612692537190083496, 3277669853275140329818758492945,

    120897095583140534663497899634335, 3918120484104979632895515090888000,

    113061371294645340311087294073892402, 2937370281097484515870841077049991934,

    69360614111493125489085300851229896180,

    1500700235907335970596131016658152071500,

    29960211273698630570609569869175635499152,

    555290308009733922178999729127004251572985,

    9606201959041534516563755333111308882898123,

    155848269335470086586721135015624476850225683,

    2381251464466830887730727637891544791914983145,

    34395521020981903041262822115645064029136956957

                                                                         n
      Theorem Number , 427,  Let , c[10, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 12](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

53, 505098, 485127489, 157930376746, 26191542676322, 2701430546489400,

    194613684190373293, 10553525899550648282, 453642909451471782124,

    16048004562728053200606, 480625768081605651349376,

    12457101578635629691296942, 284344124466989364085904812,

    5797625943988861383890132556, 106836053026588279685163375481,

    1796801970578269698264782691968, 27809901799522868171849754359045,

    398932027181315969872893777155910, 5336574263104279680161612028266444,

    66928513230230257037584066017599092, 790644883475564206418788434049256571,

    8834311143305841519010964498383939086,

    93709560391258387757758098169693942141,

    946768753538663399334509508959663934120

                                                                         n
      Theorem Number , 428,  Let , c[10, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 12](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

462, 2252053, 1652584358, 459508291094, 68450922791746, 6521499192771291,

    441627099467052974, 22779307325649510822, 939254276691106965554,

    32073402299298085389203, 931716853272994012631186,

    23512668234783668961647405, 524168345316172041737397446,

    10464344710317360979890874626, 189201878476807648184020777644,

    3127672621326667082070378667889, 47652817410797032713759132259738,

    673782472571735009102887029119963, 8894151992762769374386710910744850,

    110179999505108202958499590408226097, 1286767819242401289169710025044539640,

    14225034558903409178253292880694310029,

    149391188141462171241887157935430782802,

    1495247245309092889874467953202829232620

                                                                         n
      Theorem Number , 429,  Let , c[10, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 12](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3252, 9370548, 5422450993, 1304611978694, 175761451829967, 15531676525865050,

    991372473642455113, 48731655297299945438, 1930138848395397849864,

    63689540856945363472176, 1796063844660213978061398,

    44160819345174227761786970, 962018550738342078777891239,

    18812879328071952729038239076, 333870709881867065829959964707,

    5426587327295795376487669271380, 81410710049056854531795369743640,

    1134874174466769248129156455543572, 14785769889092028516029096785829954,

    180955193855586344065616987453539880, 2089612526147287775729917593619750745,

    22858263533110834487641317779018726244,

    237701087209932159099719844984308472824,

    2357205289994361936597649798876724983650

                                                                         n
      Theorem Number , 430,  Let , c[10, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 12](11 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 220512495, 31114055814430, 450623851655661415, 1789822771411597598730,

    2984740797670609099765528, 2635284519994936992677332980,

    1421669373330303219122899394495, 515851897484895112785980207065260,

    134774605237231214878483005087267994,

    26667024249108093225012356180679857684,

    4153489376478021929354262083612089687025,

    524955513647452741459873986738418143297880,

    55170392864851638802299843885889431269481185,

    4918548408675417788248353190236675412655534302,

    378197321740942072873227168381491752863939690089,

    25433842296268624145790452858356656827097738042780,

    1513826975383333208522880485664632669324269004578510,

    80564841799280171591818011234224855569826619273271000,

    3867810315433954455509270032899651639539730025825927095,

    168809010341174242007815497636927859328019616933608566788,

    6743642061314365020778978731071939897026602522924277577710,

    248073975925509994426165932555979989750869815406313585299800,

    8448679467995917598321276611018160420688410432441924686727365

                                                                         n
      Theorem Number , 431,  Let , c[10, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/17 c[10, 13](17 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8374065377, 140423883628393799, 63333656860912365517237,

    4206493235720817315995743851, 80566554036629874952971415627450,

    631431368660048518934919846700036901,

    2499184680539866354199106854514482320234,

    5736807774883933711709271925382172702105510,

    8414293433055895666128407441906711318350091248,

    8466050878099921865355705011917551146056136522623,

    6167036163459510965240388841542946443240046309668760,

    3392256251021254873139255533476462092504561846317450540,

    1457211664564949625785172057104398001387932235987057477855,

    502420738781477870587674166144085427938550508220494037892019,

    142218758301734707321320679610001316540555203170410367681994587,

    33684914748841845798739943730736314292436756721366662324764206888,

    6784191317408567921021578626590920567203283939449888336041585621580,

    1177987111036224437282571041571045406108332509243654790189129650308273,

    178463357869202297349655264271462220945878823875906312117160174006830796,

    23836693338007017390337093437573250341819292623055421571544451229087232636,
    283271815538195489232585385171782572257287691672977395619095923985387581\

    8538, 3019481758471913381211870222085996461242635310234102222179461728830\

    92029774475, 290773895035788698232197406822188013421619250101274436408585\

    34686805888858774013, 254603034626457280653060608680253875134998946011769\
    0740698915997413000226456867360

                                                                         n
      Theorem Number , 432,  Let , c[10, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/19 c[10, 13](19 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

194642468, 53626800013127374, 113785950941172796767503,

    23177853326268529581582184959, 1092910130317054245055832345291319,

    18421349006213143532765811818821567497,

    142974481510982702072066699609940383671894,

    601548442182010522268536119882294594957151467,

    1535524723553226962469134484166483061313704597584,

    2580276790337242400226432575228923875695390907543060,

    3035176645650596465440386529952623977504081651968245220,

    2621211578296000180747586284149415037266722815809265348185,

    1726058427057805751342476602649149392067628492320003526498132,

    893632317286755042485756099354726500795572366229439488368291422,

    373061190823013934951049984575914195840672954707423795410033870744,

    128257549284513393350672404508432032969494038826761678220526918960043,

    36967345617964129528850911263545493435168452405629108963323461730078030,

    9070099375925710490276174617988195051355673446681272771811498358997939090,
    191947325376268160342420726738160298698906369911576948935376577771521195\

    9207, 3544080900239755262423333760218362979803992904519904282631718879359\

    37923756504, 576684418901847143903083936483943816389643654882220800038656\

    73010916082398718779, 834324158711811944749649475745096261824623731558536\

    3438068008940813249284276070102, 1081709519388612347974285725856407931059\

    049663444678228126522632366785041762238284651, 12656576166705918959179829\
    0936079058563356147434067641052653421262114280764088872360768

                                                                         n
      Theorem Number , 433,  Let , c[10, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[10, 13](23 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

114, 517791666831891, 46811833331978704947523, 113774791831747753338594605783,

    36126953814899461990382602251271499,

    2953167354248479546288701128699788154316,

    89697231165583028122959755595427105646790803,

    1268476945543970099431024139094051224563495102352,

    9709991761446519760063879066933806381143509205182167,

    44758936178163221829983872529148779222527502473083398197,

    134421285690470040075418170177251233489563531458419991900673,

    279328116671842490566718978609330754407871734755029730313950267,

    421040952983448503995549818246006901999842217381410794777005462911,

    478126099383977241089581235084704808678675907672475010018087994395800,

    421891593296516785167958913375442004321377979217217863876099151591953966,

    296784037565989813541049775803447747308365484739547194154058552936129897745

    , 17006896866799580242071390186350083271252272480851951604451828822204230\

    2088865, 8085651431649734146162048229033879258330676969501640267074527648\

    7300064467209884, 3240009272032791548310818016601343544671470300986778961\

    0237989334791108648624540770, 1109279982191261115813246106586591513676019\

    0561795123289975624224621226467808106850845, 3283759272549246601570680978\

    325756426243030709058696670862704458813179538731598489551888, 84934571313\
    910665271702933305037362609117694694808189757570604023129177150095723463\

    3262348, 1937338866948797537867841029959008254499714523207173100792448889\

    40790480648660894539072750198, 392931775311651605345552533237164728558972\
    06439386384080716841049230790337020511839211220351374

                                                                         n
      Theorem Number , 434,  Let , c[10, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

592, 4065860, 3990888448, 1437396343372, 271161418795280, 32161241133798509,

    2674719088760988808, 167554136967562028562, 8312109923727888725312,

    338755611226992634384255, 11662431594294279971372904,

    346641292850695697294073734, 9051574960684874631641966768,

    210612369536209243508095721901, 4418369670635007127749666439168,

    84399752537547352110485276516874, 1480328344846817744253844642791424,

    24012263071502700970606147273506216, 362466571753358550138247957774935128,

    5119418935583448630862953953836664656,

    67977333358275892525003125681924826560,

    852182056244963666331891352936518105374,

    10124129706640228137744776870188524559576,

    114366920155822406732512998935471733092462

                                                                         n
      Theorem Number , 435,  Let , c[10, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4489, 17989896, 13820476430, 4285181446048, 728342933602713, 79888458509386776,

    6247373684219599078, 372215145169073151712, 17706933389519947471034,

    696295632858170256560200, 23240561536247406785816136,

    672254906089114162448050336, 17135865110311556183297462589,

    390202861402886898212501831680, 8028031220064675022919728247846,

    150661725797643917170082597978112, 2600138366510125829839014568217516,

    41554617376710501407579150386005536, 618726847596268241591710562387539466,

    8628438956044899265351362202078117472,

    113224617346812538556501131198611848336,

    1403834291490523427369079112650599985144,

    16506418635046713882589671588445707906754,

    184663452800685847134165920088380221636608

                                                                         n
      Theorem Number , 436,  Let , c[10, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[10, 15](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3730, 302793675, 1758070537445, 2744610910042353, 1859489519895373865,

    695719218852643076195, 165733842488350655958415, 27577547669765657855727380,

    3418541672427701231197067962, 330883935142852303082598005810,

    25912262832955566481255388227775, 1687957042502787284917066421742730,

    93505665795095603423758052014887260, 4484714189953191788643226040284169581,

    189015309047567994769185399746044788110,

    7088013622224160417546715477534568931370,

    238999134380752140391632653527538446784290,

    7311991871023436991104536536085973178314065,

    204566915624419135873831962537138832026545874,

    5269304609698454119645753492590535573913827170,

    125714980134842942998524071079775469708400178135,

    2792754465863679769770134015476039281008854317170,

    58040645945381331681477452684024563361745880278840,

    1133210986170050187339027285241405387210506592435977

                                                                         n
      Theorem Number , 437,  Let , c[10, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 16](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6039, 32846216, 32841018925, 12899137358862, 2723807547175050,

    365663409521064056, 34581401673121595226, 2467038872785368089600,

    139351469848033937349554, 6459524695656275095855800,

    252545139375494934251574160, 8508932292206404076807575760,

    251372436296472408811368725375, 6604034061786125514363693644050,

    156117922038362245655048840952900, 3353846975394214521182601474987144,

    66029459333921979176239151627552674, 1199999596411452042642697916948996200,

    20258237752151004091782970236247683645,

    319436867435023699008176097082899256400,

    4727502943111115023076553860939230527812,

    65948251564126387945695566344513265509784,

    870478814168236669563814332757950664198475,

    10908937581407690675722887464338355599963950

                                                                         n
      Theorem Number , 438,  Let , c[10, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[10, 16](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

338, 14927735875, 3406879262727330, 89354343502472839125,

    646868349985574263795500, 1935221986999142033879877696,

    3001833664448238870165315292750, 2785375896694478896353255556696542,

    1703839936339506764973624289688810574,

    736700207148650314727483383954911728750,

    237176832102664327028977569770862942936708,

    59182149746097931828560027431659637286157500,

    11814405081839304634988267436764600032948814210,

    1935723253220663588116740309924208858643323151875,

    265830474077133970502909409193471588555457197845000,

    31138322439669619852733193431896133581408400047841825,

    3157380319993746129141222787349030598850669225829835000,

    280653683196484716385795273596400324857094050142296468233,

    22107509339655329325929673029351717319640674426080961053500,

    1557862540342165330384256732415296660805522369313138702912583,

    99020227550610464329602016218452008297166472214223781776752218,

    5718532038627729436737035538873502080006268649077297338078813750,

    302005006281878056345114442175372514230498596715274145354821064476,

    14669414265162173483985195667863686671148850580294660816288531931625

                                                                         n
      Theorem Number , 439,  Let , c[10, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 17](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

79, 1751013, 3416755869, 2083164120450, 611566951906565, 107038593262598006,

    12655777652888022397, 1096046051393783393081, 73538316947632006265147,

    3981742409175137710241342, 179434441531774433764325891,

    6893482895901536079446849182, 230134131198035905120982772061,

    6780863629527061298000138899505, 178619314852850622549681477038285,

    4251907109785876404147833760344515, 92302334798234413681205764464828809,

    1841660398739692783991293945957916882,

    34002408926134672666991981783522246543,

    584349025460355520942007941307087709416,

    9396046139528219457466323542671120508895,

    142008775794157282540693819232032689808949,

    2025580610714336887351020159911745298315283,

    27367058342928402200091099488381239348746611

                                                                         n
      Theorem Number , 440,  Let , c[10, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 17](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

828, 9072141, 13267762275, 6821394070240, 1782036808423575, 286015310962701706,

    31601893563908613861, 2590536715905369477350, 166033257773882399309505,

    8646650555304790456947800, 376759486685404691223096111,

    14053702750733592332448261656, 457075004871300310131115377400,

    13156652211204989829166827194995, 339345863815962393157726957348000,

    7924933957898721154675084887790064, 169060703646534368494074538118667108,

    3319544230335111115996412518074374475,

    60388905603519921301008663232838341340,

    1023696029237191889879936171323551586875,

    16252189798669319619243966953480456551617,

    242727979833744381465248957260356387116632,

    3423908311747486343866669131296502004450650,

    45778823331210718089601808131558035771080710

                                                                         n
      Theorem Number , 441,  Let , c[10, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[10, 17](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6946, 43662345, 49480123790, 21750800427852, 5093580803375023,

    753014298633574579, 77979653312348565306, 6063098339196918469578,

    371777308352520860351009, 18643775951720175981151727,

    786193137705294410902164529, 28494842751679161676509512242,

    903391248807615865246047684847, 25415700050849197413420040058282,

    642144977925797706846999661889815, 14717600735313955960896154408707422,

    308626780779676201588081703280225176, 5965161924862256913575046398247191227,

    106949558147051001436332724454168935949,

    1788675651917808574244753973930045534015,

    28042581573831441359551618979925439969963,

    413935668008719856275376648620003050087612,

    5775170708024754065338221064925703516574129,

    76423474879575639608549624965898633911986652

                                                                         n
      Theorem Number , 442,  Let , c[10, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 10
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[10, 17](23 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

380655, 424933273738978447, 81904236235957541570238285,

    494099934214441882650631162821472, 388842533025610559618424776941314246525,

    76385806020869055842895599468938107710838175,

    5378094506599205508676502295994480346361148254812,

    170229509504940201005959232573648867689917950268594775,

    2823598397826223110107151584367432325725135359499600535317,

    27383785114363903310610446938586069198835019449600453176898850,

    168464462346027414201715456466485239896933749781742903308919927620,

    699930144443460024505873278635016362040721568527994958311738738844816,

    2063409761349133624917037025468627147199042679008355923145463510835868245,
    449133917363926153201684547028295765969886165600241158167217138489066106\

    4943, 7457337629262152196246252638638765883950074126034171673244488701359\

    268018251050, 97048112355906650332677078404346315041598740537552646513575\

    23266757283922698789750, 101276779958209369631778391271710915309834763452\

    97272397015204872227899106216603926271, 864187961866318541371237134440953\

    6999073081284692155182055738731172411313746373255474875, 6131375666975940\
    702464155226046176613953399153463436690362643566088890334619491465654867\

    455, 36700650059945309393072396003854058342479457587650956822588500748848\

    02986019816602613725810780, 187706206074723858194810921156561625069927011\

    2962864874975177380849412677048572865259726288338530, 8295382360701705600\
    284500296327124302700700034991387861459457119251807259776602531888998824\

    05870048, 319931688882664758541938207241905375166837473914985534143203897\

    440353718255056990833313158105358506250, 10863813256645315012872154935431\
    530510899233545866126212953746259277073250292474153899434603853241105159\
    9

                                                                        n
      Theorem Number , 443,  Let , c[11, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/7 c[11, 1](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2662, 2099536, 374469247, 30592205096, 1509838226236, 51955106849800,

    1358200584724440, 28533933404500560, 501039626126424106,

    7567679802028584436, 100480171206869248248, 1192855290938848544392,

    12833849161834262844524, 126520535301146591048336,

    1153301444623769374321626, 9794998255771125173189664,

    78008871595288756957645215, 585811784405163765650165496,

    4167903538913952990920868412, 28211555349915069201253381704,

    182334885765311840116478656764, 1128872127254819310645824481384,

    6714211709389545531360992632710, 38461937709196101529242946418812

                                                                        n
      Theorem Number , 444,  Let , c[11, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



             Then , 1/11 c[11, 1](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1694, 29146568, 34895703028, 12410214439792, 2105384543899935,

    213819893096948056, 14804332320913231646, 759089730597449073704,

    30502100055573078968888, 1000418963683584063397848,

    27611096358068248738299574, 656519257593223802759092168,

    13700977185594424687052429090, 254751845226245252799831303248,

    4272680178702438065411540766270, 65307520568889330429003015445996,

    917637341837836501086147771762196, 11940869192825260027516116630156192,

    144818495415947937833472916656915328, 1646031330233475097135445911214873928,

    17618990767135188455909634781210576228,

    178363437463614372129759912857666645656,

    1714175925479354750918031179804616548830,

    15692582059766414578882753466584348183856

                                                                        n
      Theorem Number , 445,  Let , c[11, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

31, 2795, 93767, 1872000, 26713728, 299605819, 2795465283, 22525020322,

    160888703691, 1038346304256, 6143547312000, 33703653455727, 172999544861903,

    836991802773591, 3840170306520960, 16793843720450531, 70307410096512000,

    282823683540624000, 1096702261691853614, 4110892119717020791,

    14932272089140693315, 52674741703637162103, 180802854717061385623,

    604902161177670978695

                                                                        n
      Theorem Number , 446,  Let , c[11, 3](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 3
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 3](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

13584, 469871096, 1144329512224, 780069782618891, 240749998679952272,

    42625857057983933696, 4968635940765938576528, 416589259897840388848584,

    26703431024973170647694688, 1367673396069484277439055552,

    57858082111399164851454460992, 2074305296819826235355178361408,

    64322573976458685890945791093728, 1753894053308538862523284270165632,

    42627639594003723406775696400026266, 934010728043408474899449757870051496,

    18627192210422304307049837547757718352,

    340905316040798565906559960826452796216,

    5766039603767281582886155594487590693408,

    90687163261570492832785349707810960314432,

    1333444604432781295482646891887498361375136,

    18417306945188006548126838596661638235822720,

    239954130776909857668645220477418710468662784,

    2960147008863701521063322649684623337583185784

                                                                        n
      Theorem Number , 447,  Let , c[11, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 4](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 12833, 4116689, 467070464, 28723405354, 1162966396077, 34556293947327,

    806659047421696, 15491057150861071, 252941095558409140, 3598912399059261917,

    45471502283868851758, 517839142544978128109, 5379623014692131215232,

    51485532739726076496568, 457670325256200089539007,

    3804989556594878936883617, 29760730479820895947122304,

    220101643377377065118677338, 1545975748725908389382200195,

    10352680990876406787072345600, 66320194041035916560744221025,

    407648952436674795388590690847, 2410668622445347339245462070153

                                                                        n
      Theorem Number , 448,  Let , c[11, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

151, 153471, 29962972, 2575677087, 131756890496, 4665960004762, 125050576884224,

    2687304579008901, 48200014460043975, 742932595812483456,

    10059893239487410889, 121735954941765128611, 1334589812016431216275,

    13402526108796679834476, 124423116829350343167539,

    1075991990682596859736861, 8724095139502975795332143,

    66686797318884999432313984, 482886004601422357925224729,

    3326164543938392945874108766, 21873806382772223644305539840,

    137780650685292107574440231936, 833644060179032427035750197805,

    4857513825169053868519674443264

                                                                        n
      Theorem Number , 449,  Let , c[11, 5](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 5
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 5](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

47, 7392, 400224, 12258192, 258768480, 4175189568, 54808557315, 610017753696,

    5925743662656, 51321636009840, 402770304242208, 2900846991727296,

    19369296433857504, 120898791827575872, 710279997049266144,

    3950385239124054144, 20901517835101555872, 105650034972372157248,

    512029936002691456987, 2386902169355614879008, 10732489511752217140512,

    46662316557428395179696, 196601772874665314559552, 804301088696283718294272

                                                                        n
      Theorem Number , 450,  Let , c[11, 5](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 5
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 5](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

109152, 6464962368, 28102068074447, 32972397771157056, 16870866360305133792,

    4794652310918191705056, 872885026813797933222528,

    111675098183728471127318432, 10706443271489627128763551680,

    805967400367522439895502119168, 49350603349868857900298142474656,

    2526243811679807824689368658949152, 110493836065830812127175473762598272,

    4203060348141006294408432715811859696,

    141089847273748513958128960825621292640,

    4230863532750925538349039776403052355520,

    114512365334306531677236266456257949156960,

    2822308329166086093920858958772115223992160,

    63826311872593227547811619798884674114419360,

    1333287874832603737368208212609035838956159232,

    25876638173966013020168331203648248237153889568,

    469014123140497594116886675246459263947286577152,

    7975219089676430012592670192709580577939357502912,

    127746815502384049534411713674260749791649070373376

                                                                        n
      Theorem Number , 451,  Let , c[11, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 7](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

879580, 80100945901, 566457583959200, 1058753584747090100,

    839794924095139941800, 360823154800388612750520, 97156916584757501070025032,

    18037705754705237098233939516, 2468080150931307779057591558716,

    261318477463546581476152721587768, 22215245563911593512926436643718680,

    1560647034892589725380595386166349324,

    92705296499799663734762329396523420454,

    4744172120228931769844684905186899078792,

    212409261132220419509770827266581091690648,

    8428636851572429079032796102598003827680200,

    299689292655001615408017453491590676861289968,

    9638107670805872244448475863504550494257582768,

    282646212774472774270989115327412492280260307128,

    7611965920594643091725676112361419622765373933456,

    189430142528092539759043791668607342680609416042360,

    4380084136892785384101355830316170037567549842160104,

    94561566088607590563499723519610819238768478126082080,

    1914424128670369943442660387553167679636395958297649158

                                                                        n
      Theorem Number , 452,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 8](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

66, 16302, 1307328, 57023370, 1665904038, 36384405870, 635109167616,

    9261563084352, 116407762888296, 1290438118125882, 12840364871655030,

    116274349942534410, 968842320584639424, 7495533295924337664,

    54247843456301864502, 369594662947791252288, 2383178287884659457858,

    14610734077298155143930, 85507327875284360186130, 479361309700199053000104,

    2582152612843824265631232, 13401046983621649973848686,

    67171211917516960587719616, 325878064239850550182295838

                                                                        n
      Theorem Number , 453,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[11, 8](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

32911, 240446976, 272189643264, 112693837955072, 24141812506612224,

    3212371412539001856, 296559214952067972608, 20435944648521138277021,

    1106636834075924685270528, 48905932237502398691830784,

    1815360927786217495146250752, 57888232228435024164260648448,

    1614622376391844664698458547712, 39974880643287505891345816782336,

    889252562999984660461839835819674, 17956539061420068698027014343280640,

    332008316383278991066180404108493824, 5662892448771855833990916093605478912,

    89679015868435177617735638620416162304,

    1326020667759240165107074601258488757760,

    18397750440555778547541360446082456546816,

    240566071141398732184407555320025890801267,

    2976117673737974140461793685141254602435072,

    34956213807309136600644936583916687133677568

                                                                        n
      Theorem Number , 454,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

18, 153011712, 5962982749986, 31711186347963930, 53510312447552653818,

    41618899717341126303870, 18319421432050080160287168,

    5183987379823853142488861256, 1027486966794810241171746348544,

    151611556706724046365394900687710, 17425285432118862273087411230430720,

    1615027571320125933184357578807224640,

    124046018738526565176151910741142892992,

    8070764821904538961568629688593237678128,

    452867656608162148871075759051098801906502,

    22244863331923996018592232371583346357794816,

    968594176976983768859483644205965103907583560,

    37787539838364166268456284097732848368680382236,

    1333030240287838341830821728875167483259292785286,

    42862390098801265049832429170684263734051453308520,

    1264975991315369167482003347923037054937129110609910,

    34476227149217741913070767845602638184710085369051968,

    872461945782709485807918918899357140786256510402560000,

    20599787063102815418792064217796699996899416454363449920

                                                                        n
      Theorem Number , 455,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

840, 1306584609, 31747571696808, 131466953625690096, 188719500424043255296,

    130734902645508832636392, 52684208155900271250106800,

    13897375356009389623276320168, 2600241980486794876396293612840,

    365542073948252232029335386225960, 40307243251329646421240912419769544,

    3603658831036635530721492059567122944,

    268158461833635829012648869368576291955,

    16962911956934323375066424559121570204872,

    928110958490902649805747887155156468357632,

    44561680256815791026009504014109962534885584,

    1900534331945094012810633457961254539028899688,

    72753586536406072668878787333968649335723185896,

    2522221371010136177502676100069622920739883504776,

    79806256452590483339102202044215152675385189990400,

    2320434309013693742232219466355958499052768647735944,

    62371039329089151026028290135884774004840737492187256,

    1558065775198083880321630240701211270936892246200979848,

    36344217557885632557987966062451779520687259545665363334

                                                                        n
      Theorem Number , 456,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4446, 3649321512, 71714260516864, 264229723795810752, 351238802201322237864,

    230160175326863422169024, 88875835701777422776989774,

    22657746754985792701245442846, 4121844270455713957805365857344,

    565887994636353874839352622794338, 61143990428599988127801702220048872,

    5370759936783538988565981662195244338,

    393477527566278343841027250942257035626,

    24547702323275336233100636337973758151680,

    1326507365681404164547509544136436989017672,

    62977967402451883723322619642772753461492114,

    2658642044719312617487133845444222458028721664,

    100826056817656692926685918301103976988240336064,

    3465470465009410310295593343448374508288706075520,

    108783068903434571143126461878701434736473146086728,

    3139707537231187224235633688400383810914365280939690,

    83814602480428997918000935531421597060279921742119622,

    2080348420559951521017172390620508284784539220452658824,

    48236430836268882831894642788844344073491538973902654976

                                                                        n
      Theorem Number , 457,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

89832, 26323090112, 351937668579786, 1041634655540374536,

    1196010120851942700622, 704223439866497526990336,

    250368138053582209456807488, 59729691910838780974546460802,

    10285941351113855887996600702254, 1348210862866921662545652024639240,

    139984608195613901464330421311159946,

    11876211316028834607423856658098508838,

    843831334028607377795411109891765573120,

    51226219593087746454942738099655730155190,

    2701104997140543734168584759745843283886314,

    125424757512774112227179857515938941393601394,

    5188933993080785660410444653317537308029651776,

    193176347976944577387496018054877270123674558134,

    6527498003155407048968688882646186068680817584232,

    201701226995968016468143166887670910684415742718464,

    5737081206437450009954183130571395539086841377883136,

    151081537483877474230898704412937987131870435074639872,

    3702586413502784131372663978695962749540927310871997790,

    84833711288571078697229554141244794179124025103184597314

                                                                        n
      Theorem Number , 458,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

356544, 68246790486, 765540101363570, 2044236353433910272,

    2188863677276713601832, 1224158301648388146916928,

    418146668555555547266343894, 96587458611564290846971496328,

    16193806897359753712440610428066, 2074986243942597235094196214335726,

    211278019516632085223932984432677888,

    17621490408516103723416624176005207402,

    1233317471009838155629904482764861995208,

    73870875026885706952462259501736109670400,

    3848325886258319810492924042907859484893650,

    176749385508399154549283698960314007504296768,

    7239663299221312347848810241687754913041125480,

    267068105846467729255985195663327785403037245440,

    8948637298581959446792871207401428012495900809100,

    274369267744424659560525262519776892959406099576710,

    7747765603808968063738674479636705003934475205512770,

    202660724685924349265196409552097776165441838143709946,

    4935451642755524722745658885038727576404610706958919168,

    112415060763063428995153779285841937103500070239433541002

                                                                        n
      Theorem Number , 459,  Let , c[11, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 8](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1325098, 173211591168, 1646296694238216, 3982283531248778486,

    3984745657444951402890, 2119386950276547817196208,

    696118077171800364260730458, 155778167747895165891920140992,

    25438560409364776476310974316032, 3187499437516146629103036872811712,

    318356723321264203138092440646066510,

    26108342287588042700607701740083630888,

    1800263869332926491301422763919526030238,

    106403063105734751065805371805469101351046,

    5477099837854949590157896349597311542615104,

    248841084893448513796033383415224304576900608,

    10092140982868848735697910109601988597264189034,

    368931066474838791874687846398561005777863176072,

    12258820448796892160305465372205801882601600603566,

    372964297358438558017499591906787075696659561352384,

    10456524569480880798516435598317461682019384819110912,

    271688623223364897776186944755859903085048371761520900,

    6575194036742353741231683467732186830914052751925241998,

    148886441175089142715638000074183345492622930815092605438

                                                                        n
      Theorem Number , 460,  Let , c[11, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 9](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 60680, 54089152, 14880844120, 2028206295672, 170466039446424,

    9995495978418652, 442046430441477472, 15548123510320117816,

    451843491244253770872, 11162799549969925244460, 239650922775005330960112,

    4549429525580977495501068, 77446943209273387937969376,

    1195999397973294788944987996, 16916631429973298913260872784,

    220942727633717243476267224720, 2683112493091186257082517373760,

    30477969939417891408756224726228, 325519151773329294770325046018048,

    3283868727469629004241384029155076, 31416349194789055787843903867313232,

    286043366200832755939830078856187584, 2486527237537852340973730230889391144

                                                                        n
      Theorem Number , 461,  Let , c[11, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

348, 1115360, 568424516, 113854604600, 12546265751468, 902520854787528,

    46913373692430324, 1882976339532039216, 61119678440062245944,

    1659502230321014404408, 38668054713680398498436, 788809551247853130502784,

    14313761238669741695388260, 234057792029803977030876784,

    3486028643219267274615985740, 47717241675641394523024394544,

    604870606346793010822494410516, 7147036347603090882473884858056,

    79161988694342592496053522398972, 825984034230917269578145020873488,

    8153946849105400772485918276780764, 76447759766729528860526013173147560,

    683027187957013493955158571036152448, 5833200409652226553734605978165693136

                                                                         n
      Theorem Number , 462,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 10](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

182, 1344175578, 75366297551978, 591556959055958220, 1455811296124735084106,

    1622837239919514682464128, 1006406552086908858699743254,

    395018404284535660101158683136, 107089940467033427679146005790720,

    21345374525508343419484343648937088, 3277174100123673160227659452037155370,

    401684385827990419360714574511300356800,

    40432306526864912528782402956433766198400,

    3419127257442762219143719966211899770562932,

    247486035808282090763044811565237382948518294,

    15573302733738863259263899745592049004871955776,

    863161261947956821506650112570503266303996779210,

    42611960470917132227147064066899877301292379871002,

    1891784029191118455126480389301223159359220937270530,

    76161272155733007597721541697127276800198926751089792,

    2800839997459323112498497031909471818241466038505391188,

    94694633938691797692324106586901644564086053992450519296,

    2960191005850052218197656446408997657666632711991449171670,

    85995748126880141167698039489831176534601671295217217407052

                                                                         n
      Theorem Number , 463,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 10](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

35828, 35089960490, 1008184935338938, 5472211311976922250,

    10560045040845308784230, 9870319034998691492909420,

    5346137575235183736889810944, 1882953156041363157196665623798,

    466867776177939604447180594051776, 86308994842024327919953096094129494,

    12422025023835491189784233392247420620,

    1439279534227887858367793731897324713590,

    137866489844602085400317541758018235280922,

    11155323987258561149820237838195195648920406,

    776094415722791626472705594039155989739307008,

    47118016676558437159589102996165637767336443948,

    2527773011751678111481923095716006161671985046528,

    121120157804014688847982985797636770778848920406528,

    5231583151851651926221849813543513117339889706994688,

    205342148889507797313014381573573590780467754263345344,

    7375823347399184357117704205912398679861721095501040000,

    243966343090487922063083262134948474652833939624946610646,

    7471933501337639281558018805143831661376982875472810615686,

    212940949908202626572434685517092556092304608869615577066506

                                                                         n
      Theorem Number , 464,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 10](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

708474, 270243381642, 5312038250859392, 23114437789190659820,

    38406845722816708138490, 32155361088682247175611904,

    15989154731044803545422669722, 5256324408949078374235423418048,

    1230878966404847816053997730816332, 216799257498101201043404290428542848,

    29928799022739261474923759306833268390,

    3343730305398265753800922017183943041024,

    310154737407842485288488679144224319617530,

    24386253302473949599447392660147259120400472,

    1653388843891062509722182618881687365748180442,

    98062036094461455927064593326190731629354913578,

    5149966067729112553367253053210145877061847977600,

    241995895125420293551077077384966264120696251588992,

    10266450655186837855866392430758114454983766936616960,

    396320792045793654694652663750519072591306319029409002,

    14017693884556613295546662221372684726126918888492635328,

    457035685389958219873168218696961211590260215805398410870,

    13810654275625019143262562014758098179342683249013882057216,

    388656445859599831350360658841214713621625957568793239229440

                                                                         n
      Theorem Number , 465,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 10](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2827136, 724694860864, 11972215522687572, 46946548881410959014,

    72627659134735173730304, 57664986241418020485276748,

    27509949709251703917338417354, 8745316391738949148523182023680,

    1991547987209917983110306611289344, 342566594392172233088333098246227206,

    46333679633875836232634412285178556406,

    5084766845850527282918724012360572098842,

    464244683521482127599220274803777082613140,

    35989650584806113060638695183988770742894080,

    2409260356840298267685093082226503101130006528,

    141254394372489244312935598533238816334259789786,

    7340705466444857078530576823004360240817293174486,

    341627339068141186795858683921651653227772928845772,

    14364980964884390674042728723367288112498391198034358,

    549995956825375901301708299104157937548673010910657126,

    19305038271813966110184227089296432593791178402763014144,

    624957408638538300070221255885423440695689508583968719488,

    18759494807097131687561702497791401326933385746997266636318,

    524636934218120179066850575718113020105721135640540150267904

                                                                         n
      Theorem Number , 466,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/11 c[11, 10](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10637094, 1903333972332, 26674849789537274, 94634303270853602230,

    136593477069550684535578, 102981154032542416695123584,

    47174355148624628499562082030, 14510326152549646920466441476726,

    3214865286806548046103169050988480, 540221624048363196518171448881619178,

    71607070878850340485513945310263721984,

    7720614947449312352337311381323063406336,

    693951857155778639063428031916005486560896,

    53049870800255033655549229542770661128445418,

    3506849787292593077369657094038066662394182016,

    203268751748921253086399519990649171916193787510,

    10453833962480479234497318334807116150536646000640,

    481873918562635349607948934215517373410872878209002,

    20084162045512659494541761707619144331325441055112634,

    762711758283457577244020070192456325212627550885874048,

    26568963774825720029991572286692370511421207116999240730,

    854043022733963613919614578839044671988093365131608512000,

    25466788154487032591249233875441518133632834989647700819968,

    707804619503511947355219706243725288864485714629600314693632

                                                                         n
      Theorem Number , 467,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[11, 10](13 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

154, 10301847040, 1970524154615040, 39724002899655426858,

    216223582405498879599514, 484451896174293532770435072,

    564197506758374229151447413324, 395041964458256588378383579582272,

    183445525575316400882880553439536256,

    60590598435950288103127184493300072448,

    14993909256884182087773419911387397694848,

    2893107679374644954660070740640838267399404,

    449162489281459712208034387249672622105763328,

    57545735244460511607888796420084465521584936182,

    6211366163914879066448949619465227896076171147642,

    574648686758480907137350848968351669162657724108138,

    46233412131116228001271764112123978411552476337258380,

    3274995269085234256453141973266259462766782667004510208,

    206433059538105164822608914114187170860090256605724055162,

    11685938233221170044299090935564544613289962749165592509952,

    598911601118356263146724366821613705962257143148123343202304,

    27987061310924800766332469252036690160755546402497008859537240,

    1199987085782001880992212102537619992625418290421852656334683776,

    47473609051555595782102591880225856956985138662779168044161454080

                                                                         n
      Theorem Number , 468,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[11, 10](13 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1034, 29691505030, 4494801596881024, 80075179690722278810,

    402421692621330406933164, 851574774842769034284398138,

    949506032552617145345781380096, 642200808272163489803560491869914,

    289864041408761120305512621592961482,

    93481339844421679955703796382018096640,

    22665314986385429047269606194231334815830,

    4296389716165717627080965881915218269385574,

    656695274842362145476904733417747375933259776,

    82975569003005847322900809737546003686377234566,

    8845551814406559352266961667913713665838700462080,

    809214380588823556539351389920950696486227678538752,

    64444153362543314121148996820646157292476014943229824,

    4522557534882825025769746060888427790462474499880761344,

    282635363016191254601083942408141609057890878635273547924,

    15873418682928342197167594421208108815097480247459225615346,

    807569899458260464889843825073719267772907233633897748294454,

    37480628900732826279654341704053595116303469246548581678900288,

    1596817640655227150310146238166678257000056021386522239469338922,

    62796612345897970009255994277186901892894476054505496690872456832

                                                                         n
      Theorem Number , 469,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[11, 10](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30316, 228667476774, 22571026744993078, 318403966318723563520,

    1373169972239589346700416, 2602015170412588109260976640,

    2665792638274809433417413603466, 1685155989177622908785644055706368,

    719447798712681102759370610046877696,

    221401690396969433816107930620177688026,

    51566763831082791003352730237220986112340,

    9439120296911090203694047401549780933701882,

    1399021329446283662072616062130658540248460374,

    171996636095241060303876516915164683929087050970,

    17890641760102621489542125682314550963189160108618,

    1600740332392484846645483406331804211765494639228910,

    124930093719214977066679806976313822602277900107996672,

    8606762309180052926466175687461610987617104977665202010,

    528810115001840100059417985749204449819429584186435070336,

    29236418155951262811927745901868587306979854710444911904954,

    1465914843420992508008633844314095241173924818683608562755820,

    67119331786293948746261972579594507270188782513980911880067648,

    2823560152375963834459333235980216204318765813185821620542578688,

    109730297376870954230287739089896556630566536698960524185183451258

                                                                         n
      Theorem Number , 470,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[11, 10](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

140333, 613203343808, 49740008849844032, 628360575993072204032,

    2518224708151161455782080, 4523654871352847777368301568,

    4447659115264604778199204430656, 2720270627297848346702681504682560,

    1130158541649676457685379797089926400,

    339886788008299585612912332278792609600,

    77614783019247089055401289284327893663744,

    13964679525124734073457958536764103843566400,

    2038604917326406226502771069576271854802313216,

    247263364253694464626318447379295378962486179761,

    25409443555186267625088563666891034520828025072192,

    2248651444649767460675966733903519388544308649176256,

    173751049060352751572550630562254576814241945915138368,

    11861125594624826634693329571930733229799700597955306816,

    722651788467199981008904643633037799374540539726745664000,

    39643673764903904887157437258113938813893257445235948732416,

    1973433729811812653326673437754124550363909326274001869246720,

    89751374004617434581000776962482315874005193713669179141357568,

    3751993448850500455209556368508121495987941343508293390909944512,

    144955525844675394807108448275160171288048739684016681848980971008

                                                                         n
      Theorem Number , 471,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[11, 10](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

599478, 1610513361204, 108463971936116736, 1231840327490160455782,

    4596536027085791311419392, 7836723950865264148755811130,

    7399883100702187741058077096960, 4381222579307053118088891015520294,

    1771953674342081755964765913087721590,

    520933490238689442632428361753314859974,

    116656260637740226107960996872169276006934,

    20634522025170265045686255327816911563415784,

    2967334435401424911620479079570671791256615552,

    355119311408523097019028948671618165917086212992,

    36056274244622188807585977287376819254939706044694,

    3156277443098563903861869166230198689814686075518464,

    241473659432144680519273844415096970365906283559985920,

    16335032383842586708617422921712366040900227879261011968,

    986936169030525166991927336382112484804125524474530851648,

    53724716709995738390316101634507568844692843101219655745882,

    2655243676993990950720882318805959478472552921389290544263552,

    119954949448152956467170859733729172067127853054951396797972480,

    4983386419644543612152691429622331764335400218572854387560940054,

    191404838955410026968217958222119958536790822428654720063709885146

                                                                         n
      Theorem Number , 472,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[11, 10](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9000618, 10490514452096, 500548196124272340, 4644384575065060549632,

    15107404437020619221749482, 23277649753982211006978660838,

    20316829416798546792828078028918, 11288911230675485285098950297359570,

    4331508579236257340464577948515532800,

    1217851913577443572465056234682351680730,

    262438623960482102936413497737420578137910,

    44888352215600413093157040382344405570223046,

    6266486999140366364119077197939635198124252820,

    730367221648271156659473172175041225334151120870,

    72412456520156823676487838423741230977462343642624,

    6203638513672756796208873630728689595594775575322624,

    465381194236856531870676253088133639464261778462863722,

    30920007362015715511827515309123680752473750179689652236,

    1837415865896158613440430550055253195555632598036647594250,

    98499003917316556568121028422380519977688356084519834386816,

    4799292161474449867875140291664018848415839644428207557956054,

    213957513796483808194622137122788932204891435286255863817943398,

    8779018467903593398773404996714210380786330617677347931702647724,

    333290206425413563352847979389459775218601890010625809209304476346

                                                                         n
      Theorem Number , 473,  Let , c[11, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[11, 10](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

32164980, 26076352251530, 1060132282276120448, 8935422726869107432810,

    27208382459654209148594688, 39916762048836224115014069410,

    33529888220167091800090251396986, 18061470752353213662640598472177536,

    6753571448242908696348360369986411008,

    1857695148099338245412476338096404248704,

    392822424518177184891647924833965223749580,

    66087523136352325704034151694604915208974934,

    9091939887693621957097303825281352842483580928,

    1045925034733105195752989397648081481513395551788,

    102486287372627813640600987982615729120564471113216,

    8686996708235016647271562826026096846524725869445120,

    645371487778310257668016982470539967487607927672662656,

    42498023534141477249453299571195339880823684442133622390,

    2504777004950044184628786223884536479564073833223533914490,

    133257638565824753520646852331459646725107389767483999626752,

    6447192650785957448989338867897847597629495972926373147509004,

    285536785235913510562028057891383034037989719086738122006285056,

    11644124885509957078879071195978454089574576463552896406700312250,

    439517166077994021772213756126919716297192725115675267439033114752

                                                                         n
      Theorem Number , 474,  Let , c[11, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 11](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

88, 31856, 3573944, 211237936, 8167663328, 231803032032, 5180224983824,

    95520753424448, 1502190469380656, 20646222151615632, 252671752140583864,

    2794124472692933600, 28250436073598389552, 263681907302166224256,

    2290275656165602585360, 18636892122202660085120, 142898828494274937303560,

    1037517600279226151297872, 7163616828820976214257248,

    47213290801559418115373200, 298003996374279040640743872,

    1806648548930544226194397440, 10547485824154083241217942304,

    59437442110986384018636256832

                                                                         n
      Theorem Number , 475,  Let , c[11, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 12](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

58323, 111094508345, 5438441636205456, 46811441844663266749,

    136621121922862662059110, 186571245085568527606049019,

    143753059052909508695827210117, 70492867132870968116931659455399,

    23905232219365958272314094191993698, 5953814263254727828229627421000967329,

    1139521570660460966821462728306197441398,

    173591101260623161980223927127870166122635,

    21643387339796854264968648399461967062919191,

    2259082365299257625293551107350823924119278390,

    201110692543805116482527471495412939025236741656,

    15509314833373076650011907416313085502150951948336,

    1049835975976484375790251962845882855882087356032551,

    63082650573836463140597539093878995187705505501017819,

    3397637907366466924767404845205158176833040269610920559,

    165422578275790710207235403929463085865075312628543429293,

    7334701717278575217709319813872714874101419113412774742651,

    298113587929899215191415557258890010086774020555818162944979,

    11171618169620046107531687072804375817380434723136847754575682,

    388005935542107962361001841290605629391116960070151849046770937

                                                                         n
      Theorem Number , 476,  Let , c[11, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/17 c[11, 13](17 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20769567, 3109667102355312, 3830124748480731608282, 483422577436489456664057920,

    14818990091257752803436387097625, 168399070316450648864460295412149456,

    906772036007810467505813200143474630951,

    2709113639239787893698024020443621215875328,

    5006443784066500903856878293134070894977862849,

    6191284338410035347329355466543213958878043382192,

    5436054749101481846552506108474477437577117153988301,

    3547759540325946875820505809874627262512454619785848080,

    1784844986107231167459704300902885155063473416998858306437,

    712892375624951081429356386843403976892955332165012847268080,

    231614238230241309885873551251209691175560646939500273038650500,

    62463784641544241567072098138442887263969378457000121911686095544,

    14225167145128295372975448678622962147227792858976534606158219475944,

    2775970774691126502349419597997203434427732591803759218093853449773872,

    470099719546141359440434224029440554990287257577700806302358270256099522,

    69848672939685320587610404338612659837577511269172634816744535104830654432,
    919401109568867618803017535988975971562940209529394984699107935550706325\

    0518, 1081244337639234990533344470671754070101460481287226192348116728576\

    000664174992, 11447039358944573128231240544363640275262649731950196881084\

    6222404307649670717195, 1098346604460261958509085555993123099508614527030\
    6544568888159712773999186931918576

                                                                         n
      Theorem Number , 477,  Let , c[11, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[11, 13](19 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1025175, 2782333723160016, 15004915560159125011301,

    5474057821026523547800319968, 394914732887233438739994299587554,

    9320902091204688356225027248198929712,

    95741391378702392248862098835613325668115,

    512716485350716717702610635746550606200300336,

    1618969868651411540437613355794991154529938496580,

    3292840618387814435845139324365932847443210112436224,

    4608514540930579200335203029880184969479380309569885625,

    4670107674923199348935498394897125666568253584145067994240,

    3567407107618915591718968532333313812386618634797976057311525,

    2122010403790679242706876998740826345890875450023265822126349104,

    1009474212692693997366667866266088051266913276584767167724739393944,

    392687215442510412046988318248961958524943280630200007842899002518864,

    127275485219320596684829912342837396728385420585079613074095654378922501,

    34924897730897933429708986918603348406177053251773233762699956097397124056,
    822622045403723658929015690095399553503683345368406038941307100229579343\

    4674, 1683205333652092981663594097695145049361800821121024331113468533792\

    572274931920, 30233893471013735689567221355301475942358260569385019027619\

    8033504225405346508375, 4811494054729535863503824465667015798405086413360\

    6279474854464314002117769839027056, 6839892885915611911637551752442432546\

    587623044823729234563589357546896664793724562066, 87492671018333850887290\
    7278569193012833251856299907451706211319656818201944121672722432

                                                                         n
      Theorem Number , 478,  Let , c[11, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 14](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

113, 57060, 8594130, 663943325, 32892653331, 1177491926190, 32770416807619,

    744536863826775, 14295673459293345, 237995527743780708, 3503515147066272501,

    46314499845214981263, 556673376414112117930, 6145680885509739933360,

    62848802392864869640491, 599617566992312852742377,

    5369543588897194238606367, 45368453819952449037549000,

    363323118312258478430927090, 2768708038893964527569305608,

    20147635532601772039127119188, 140436309491087366744094570535,

    940249088022434911214223884085, 6061637223454327551876805684605

                                                                         n
      Theorem Number , 479,  Let , c[11, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 14](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5, 200539, 398365995, 221104094076, 56848251212710, 8577404075576007,

    868630500896075934, 64296300059686906015, 3687408531305843960016,

    170875101812255652265538, 6602825680973894857070982,

    217993870987355087058556254, 6269021511772045795414169820,

    159504601425045831309524214519, 3636982334072596531126083410763,

    75120675507053420670115113529713, 1418286518045329227317722687018843,

    24667107835146491143939283349672366, 397854464868267147273751213653666515,

    5985589083289196321586077715741287377,

    84426661489583146325273304458348577990,

    1121491049710704985051100872405366594705,

    14086074324852481910823520023368977925201,

    167884540048384382976065848938661412313159

                                                                         n
      Theorem Number , 480,  Let , c[11, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[11, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

660, 5156478, 5596925005, 2195611632379, 446722118296065, 56743059959726570,

    5026835931376911363, 334004025848467270758, 17513174145969709084979,

    752173910115571854855657, 27221072291971469422529400,

    848655016259706460381113033, 23199320392004542086724388358,

    564149452813460946728534330451, 12350021855612864364605334203247,

    245834647159807009339326990220016, 4487576518327306286889088644717126,

    75673071713202458583761127338079011, 1186243332408493383918152303063769720,

    17382108149786424039763737782549727657,

    239238587362770468320591160465532776091,

    3106143745497187940601890487498901340621,

    38188267673522784677456437962797840858979,

    446106463596918788396865602232828507861588

                                                                         n
      Theorem Number , 481,  Let , c[11, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 14](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

482245, 998005287445, 64907871202849284, 759457908394774062370,

    3001284400442679480486810, 5494809690709574192489214324,

    5613646298005847438456970092385, 3611078226219688486415093833270313,

    1590432620008962481812444483490347567,

    509768658959411356841409487456984815775,

    124515718115608704034330208772668186347190,

    24023357755727445103564037953839260488454145,

    3767002118101484480580563508027393473371731511,

    491330558048867642156805107541061169842604589632,

    54334150990871936542653429587842421260444420170780,

    5176597394013821663591318522747509788144849563065264,

    430701754163170476366354570860540675765579007455378770,

    31659962756187000841793833111611955095019517175106296477,

    2076830088465684662668146054108683548177330486895775556277,

    122642900180797055128279719992083455591520720205417042875890,

    6569982470792641566366229360743887628703541551201621650979050,

    321445571993552831598892883333165847350787781057716811135886770,

    14450591589893059222743195394338387168294472366359438240360454178,

    600113979348322667732863439445526161583810515068771714231648458455

                                                                         n
      Theorem Number , 482,  Let , c[11, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[11, 15](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

176742, 5989417412, 24822041544599, 32511010675790736, 19882900907656363894,

    6988629607852338391094, 1602452400345501580754547,

    260758382276893119552502680, 31957648061916290863596101330,

    3082127228411513004376397010328, 241888672645747607091153400372490,

    15859331258446018395855733423031464, 887185677710049287240909303970187350,

    43080877358196662495866934591062979968,

    1842050004305215484647688057576771639784,

    70191382636281620057242075498322784275760,

    2408077660017851489737258113079419549685714,

    75036948319661510606975993848405166363127852,

    2139949929744262149911466293665851076144591131,

    56226890912296141578548916906723554917510583604,

    1369108380072040652945772187245159488740195200772,

    31055259339077598017802569290874276805319821022832,

    659234802506035904220981889173609190184216383855730,

    13150639417629394605128853511745316962521516807185304

                                                                         n
      Theorem Number , 483,  Let , c[11, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



            Then , 1/19 c[11, 15](19 n + 16), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

93822907991, 17446066535439134952, 47283986317682941562843892,

    14063273492256338600540582101620, 1004914954254770594140571344572055011,

    25859983076524807389510238341487341575280,

    305314730461089109736039332755400918451058144,

    1938404214992446828157603351866713036685002834260,

    7394085858954244526163919905094276592020530634892094,

    18378588205616098514184216088311999821679100755093387912,

    31655645885245365431822807141095282901029244323777867853436,

    39641077706821801684443401028965984026180479028348102100034632,

    37499896004628607431164053755002656904864022818054940554449079165,

    27646252094122808048795470963057166291033955685284317121725065268400,

    16298536781016674832605702989480953694135337976159814721615297634873436,

    7851264826413477299063321616609137954873632242382380310192259788248565742,
    314743672820071252451358209194547367167868291802269759720266993609478553\

    6628, 1066621789735732100896053896061875004417653553024412095584102699517\

    049918931240, 30973341605776067367372991313730230193952059031591863038847\

    1725871402469148701664, 7798705254983336940953058744043623237433503249618\

    4610323492673624601297044118218248, 1720364437526743763944650416029040215\

    5196760592843092602103510031524080299064651888271, 3355558347911166567888\

    321977799798834723447825504349266256598165247943867522922372019968, 58344\
    011679144691741885583919189945481896913793549225707478152601260042493934\

    0943148656296, 9109185527068234937137348327436938022470486190663700009561\
    0150541109310971566020198412906152

                                                                         n
      Theorem Number , 484,  Let , c[11, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[11, 16](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3979655, 8859006346422, 740593725740776101, 11427221965904034889173,

    59518827188716658870163803, 142582277683772101558837556384,

    188908846870783977237715362329033, 156179858216606530301958778540822295,

    87651243266209233437192459396803794901,

    35514129351999518449006593553526121977781,

    10885138636256949164273883049592468463853238,

    2617424365111348525054069182742003337524665506,

    508331416286090067856176654244606275703752112096,

    81643923944638440656573869159929598544037571182598,

    11058518434236663843169908824430330599100886227555269,

    1284056724654633917486104566337971637341122689631372695,

    129605446388148757201850804587734476825317390826240368068,

    11507701603549474782727977705024241146706511273194145547729,

    908146101099134980204347967795917906632718930510404237695909,

    64272772918308065678783359942325084418367878972823970591980387,

    4111780677284044013096683611201167660196343377674225597768625695,

    239438690532395634362513623469226482315345302807436686294911548162,

    12770693782265382130073340973306592809350428688527847885445798705678,

    627334445319248584055627647937888587531527900240694961518971150368765

                                                                         n
      Theorem Number , 485,  Let , c[11, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[11, 17](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

141, 95676, 18747921, 1841703036, 114033498075, 5032821884220, 170767270293216,

    4685997037938936, 107800318169076087, 2135216376127970616,

    37166518472146997154, 577760991987859149108, 8125765830848247218994,

    104500266690800519010204, 1239795261812998852846860,

    13671043418472850256978712, 141005113547390637010371564,

    1367818246071945064400166804, 12538589386009403563849709190,

    109069854467718143169373695672, 903631198321778897755306300566,

    7153566831082951559886224758320, 54270123169200416779587905277327,

    395582529149439750864108643969236

                                                                         n
      Theorem Number , 486,  Let , c[11, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 11
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[11, 17](23 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

80042, 278506483842778776, 84201004840862382444258075,

    688027138144774121967577999397472, 686339969238049382508626062792497239211,

    164462271333317094045660618513240454143539376,

    13770750844324372173111922414313842358584222418575,

    509048287144093121089282148402463027713597333506949616,

    9726635975748930787788183309261826781331088319758691506467,

    107497549489644644757665847102460205839734486304169710647177144,

    747057312740529975289446088196057984818408068890325866114774625708,

    3480810753330492721183789503236804315248075464051031393412285205111672,

    11437095631688473955442558093346859548875621936859089638393880072889569299,
    276000661320626695016056584328329330597427275647857823572804750598071770\

    48456, 505729276788817367820673669080550760277583474342337754562776088821\

    75001795981536, 723363350352413886382373351053777644594059719403414087629\

    92524421506876931407551400, 826695135617949819999896686978810821361773393\

    69124629614370167397676045309979974644975, 770022653129523660551640704774\

    40118719681823181949538675878034838663051808784212894898340, 594625557209\
    860186836823998354423030409346894109854743411252534962367196421402907502\

    86568709, 386364008803818335337549032501743882267364297002634970953131219\

    96410717002429787643128778209120, 213986092051814540112632841178530490510\

    40821796702933177218910113137823332000118086528681241413009, 102178727931\
    119272914534330350409449258857710176835875386076710243769325664774026523\

    55062006698696400, 424919853745909140729798540974293473200881199054648196\

    1113815799023336639095066324806008771206769649723, 1552860421293965002718\
    531853685152025831753708731919036578550418379475392898269558185613276098\
    867033162016

                                                                        n
      Theorem Number , 487,  Let , c[12, 3](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 3
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 3](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

522, 2562978, 1451289145, 292802217348, 31090826649492, 2104710974705574,

    101540554591009800, 3750499247519720082, 111422638830229830420,

    2759365082913230002809, 58514531413558808406036, 1084865730837799851336488,

    17878015625321923675756062, 265400108976913126320329892,

    3588411130589828441290170178, 44597947758234147820169270316,

    513468888413366854297183324515, 5512947840742387706255687421416,

    55515176922365045569462913938536, 526940316931121857609973242771254,

    4735099824611245050767219124597000, 40437813892601687803658640753996558,

    329322752698457815832566306071450894, 2565407560244202779726890005017955524

                                                                        n
      Theorem Number , 488,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

28, 64000, 18344974, 2044328608, 127739179248, 5335938720000, 164839822856758,

    4016745384121344, 80693501701387572, 1379755710843392000,

    20566096411729799910, 272219492288832594048, 3246901962254031872000,

    35314155916714290896000, 353664641753740329347664,

    3287987927195403062016000, 28572561964451499300194192,

    233452031410123721642496000, 1802492909020165394957542018,

    13209551098703945335186592224, 92239186129570132674305338096,

    615789860185918881347207680000, 3942300528251525065020282556038,

    24268132789386968671086854144000

                                                                        n
      Theorem Number , 489,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

938, 717248, 131742728, 11315856000, 592556874014, 21729659904000,

    606634686857332, 13622340523904000, 255728630756921952, 4128592732922511584,

    58568026037441121160, 742426676149482688000, 8523433180685707002010,

    89597258473340508839424, 870212926159448325300868,

    7868730374278696730752000, 66670061038989861089280000,

    532235543988388829617639168, 4022537765384233982786864160,

    28902187618597587400686384000, 198146641320328588650278276866,

    1300394106308493400900165120000, 8193072878877473727677728004408,

    49684509454653026758456763264000

                                                                        n
      Theorem Number , 490,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 4](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20, 512320, 600453199, 197990401536, 30808948829800, 2869103845800960,

    182663307683515680, 8643905524619520000, 321764841289454788444,

    9811801986543503443184, 252617601252671663819760, 5620521695913354807680000,

    110065592832334138359297400, 1925327564765185778952904704,

    30450636470411216202262016800, 439849900132799200962291200000,

    5852194913483909805484091431720, 72241029365308135723095558620800,

    832542269828078975191822013808640, 9006066837215710967052880982080000,

    91880969456561305068695949544468440, 887743576281699733229923778337914880,

    8153146602512927726389228249269476800,

    71411549689631390877702657464513401984

                                                                        n
      Theorem Number , 491,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 4](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

128, 1577684, 1460234720, 423254910010, 60526582400000, 5303248488292352,

    322226978915840000, 14690068865521285650, 530304768678201920000,

    15758542633520001617548, 396837836512294714146816,

    8660841765190456177551426, 166751451007231229744640000,

    2873241260988738559133474400, 44831932143005483441798629376,

    639712177817204572056021998218, 8417201716057159252062777680000,

    102852070442676114793756934182740, 1174279290106221807378812024639520,

    12593503979206729073440292051211082, 127454702213785674003463843214988288,

    1222299608496482693651877377305600000,

    11147808591568820738638746873640960000,

    97005953360015288287392505708145920000

                                                                        n
      Theorem Number , 492,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

670, 4640000, 3471659008, 892157124992, 117742730611970, 9730243231360000,

    565123764946968272, 24847322500906123264, 870551789819280311334,

    25224397083367350640000, 621580661542463089500620,

    13311746581489852806351136, 252061011807365885007442944,

    4279190626636054094693760000, 65885132949692951910218098640,

    928852933077495286357260800000, 12088196676498851177177119638802,

    146231505670804732316277517329152, 1654177945140774332095816448000000,

    17589148182169927610340750800198080, 176607520896815496084820976100866990,

    1681213961814676009391206240971200000,

    15227815633779265799474679532994468872,

    131655159053614426580493490676294597632

                                                                        n
      Theorem Number , 493,  Let , c[12, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 4](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12312, 35690496, 18456012946, 3811384800000, 433310490612380, 32070228378778944,

    1709336514214654976, 70118964322464320000, 2319215687324308480000,

    63998041766671792028160, 1512058706739247194769942,

    31213146414691639733594880, 572131389071809095312084556,

    9435393641931389525133280160, 141533315228806134750198769190,

    1948750596563596846309842560000, 24820823755826409831926831808512,

    294388911329475891320989644800000, 3270135237328642856276916131100290,

    34191912344025243281696900226560000, 337990267718417003208731011222620428,

    3171000249814882810728166000961912832,

    28333531245730968091581792749665582900,

    241857373090765506339822127157241280000

                                                                        n
      Theorem Number , 494,  Let , c[12, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[12, 7](19 n + 18), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12864088090, 53528517985308446, 8089085508272559897044,

    208960144782249137017760813, 1713209225399952021786798467636,

    6159058093812743034879981419987268, 11793416531835474585395024437473042344,

    13669599506670423188207602754685439453226,

    10489615061787041325776816153539446919583476,

    5690505876095946897932951664980443236820159142,

    2293821119831648360329062202378129887592201396298,

    714276672158222308293053673512315458447220855981368,

    177234984305698059802656425303094701512444659776259154,

    35939704049529792316257774363942255575991302713706811576,

    6081452204341943074354733902794098933984785386800559200446,

    873867289739523677120790398394689872569084500641089414773084,

    108226086663797052711279493377628995635872599632373945169387252,

    11700032102492311016926514224613304429379732311865883943474143804,

    1116307586796246017915414824679837248206187336683031309354538451688,

    94903648537326732706094534267771428406308795131141177529289241892234,

    7249982341043088333546292525127164617164299355470362172969812613333226,

    501388041762734082222226574423186274090226645919592124297354027292478664,

    31598274036399943538394593619164940364506392035288911697047905821114515073,
    182544776418914672508988008026877748587041283008444675818537659869279448\
    0050

                                                                        n
      Theorem Number , 495,  Let , c[12, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48, 349008, 260575664, 67225708608, 8974395100176, 754626315723616,

    44791180598123328, 2019232008153992640, 72715116446637596256,

    2169624712454250014496, 55133619500007776233776, 1218958208662399372797184,

    23848743562092440623330272, 418615176916143695724187440,

    6667392202549301078663347680, 97274814787336992819426590064,

    1310472165529538019089066992464, 16413997484985676931602969575200,

    192278048865508776369125250774864, 2117439041281570730303694980493696,

    22020140022190381074157636695795248, 217114870853863931147666468200017504,

    2036846005360113054120242281664998464,

    18238968230453890039279610148812980752

                                                                        n
      Theorem Number , 496,  Let , c[12, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2640, 5839376, 2648251488, 508307143824, 55421913230000, 4010531145377904,

    211704566166174432, 8679047305515458048, 288794645472297655680,

    8057412192558547802928, 193219946849121319351696, 4060596040294065247590192,

    75956437148396790750452160, 1280826170624482768019473824,

    19676049372778127308483955616, 277810485009280821813175353168,

    3632345953070292597746279537680, 44264647006255656092650237388640,

    505576391596102145872003308254064, 5438729920482625030836444284558016,

    55341951114071357079740358264067920, 534698241600540329613816445751522160,

    4921870763593147105875564848991576816,

    43294650193049674848535034337867678048

                                                                        n
      Theorem Number , 497,  Let , c[12, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 9](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

568128, 1352186684713, 81424833140836320, 834471043251634620960,

    2830507640484746318715808, 4412855888319929754699960768,

    3829009337008653059098925467968, 2091819200185625011860170878676160,

    783402309753889636200093185255768640,

    213922259447185980533128604380552861728,

    44617142786200360603796409181114059510400,

    7368138074203640978712041157298672159135968,

    991384992197463698866856378703563259577021440,

    111228481888022067909743353424939172761622208160,

    10606348896528454884153865768981884803595726572637,

    873405588036214311943878741526716540292856826466176,

    62953863005061619824664617198328128822382205370047648,

    4017828789750985123698564699461923427761777660253096160,

    229320625439677227195070388578709347510405887014535533280,

    11806915482942735702270515182692293806708563789791519385568,

    552543711092949807795181118368857392664642473957972300230656,

    23661465439382215587815583201341823709612375906683123451658304,

    932702593267501209399442792599417354920302966358845739012965600,

    34023261845588656248669112598375362410333218162200131947043031680

                                                                         n
      Theorem Number , 498,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 10](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2247, 83865600, 249548535808, 214210966384640, 83919879166361600,

    18868367722365624320, 2785503257498912241664, 294621512627716666212634,

    23718053818035069983508480, 1518647475571737947504640000,

    79951733292283855296557066240, 3551489799487531368926050459648,

    135873901190940231840429677090816, 4552545576213363614264854266654720,

    135440383969523027527966654981014641, 3619298710542486775885532653563412480,

    87725111331680207078860625549395486720,

    1944808601923421028297025086954084212736,

    39720950681674965887767398302466342471680,

    752115187499885987942901611286905297305600,

    13276039777428232929077208994324673763287040,

    219528030412541495905815186576379819986104694,

    3415313287182492777542750309945448544253675520,

    50184727213048844725946447441686716542063984640

                                                                         n
      Theorem Number , 499,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[12, 10](11 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2, 186195520, 21732435228050, 263669842414329642, 887816897150655844258,

    1267707776810349658295110, 966412029909287784775680000,

    453343501762677690159203427702, 143909914961733410333603716931518,

    33068556030496809288876464487798758, 5781707984874177701855046328133021696,

    799061902783575001005942031927655834944,

    89950431171965745056527619309682535750498,

    8448388949272511863958223905479337849364480,

    675203553118561228922638320659210487545960462,

    46674198830985211514742880786419019590554917238,

    2829206744660840393996017748755845007715101217536,

    152150074751419048176846343246407553201109889917836,

    7332620271017541491050837289952226385934330265976942,

    319451649958753638686177639210654975167867461857956438,

    12676867075419306023762160915120281264024413239284346880,

    461301361426001441997360627699022931056246861362001514944,

    15484449071750303349046808818774023323188995454598104420174,

    481987227847509216898189697529582218588184427833834348049418

                                                                         n
      Theorem Number , 500,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[12, 10](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8478, 18913848582, 802985208971790, 5698052039138020490,

    13557798235154343505920, 15093306975113226353141760,

    9515497835057305407628276718, 3837119945756279443684235634496,

    1075772907572187626648791754421026, 222685044996868490902656050741866618,

    35603666738119390047657455575785006642,

    4552654528898496707482279782905383341568,

    478618755681745628690937389000757916467200,

    42302112323033522739311488918733710408151450,

    3201486871201824370610595033527504181318123520,

    210672553363334029598215976232506386509065169940,

    12210898196615036588960949345281124675203613012096,

    630328824267116298405430759631674829090688245657280,

    29255111975954487865873700440338465213004473080871950,

    1230967448525333213942794734085612244496979207038638080,

    47298883109859261018870947440835071038948544696470621700,

    1670291004446179108393729033029202461392002173553252942886,

    54517353609981465884285573485491772419329102287379803570944,

    1653017980531923446315638595948144944191344743824932100890624

                                                                         n
      Theorem Number , 501,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[12, 10](11 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

44490, 55531096960, 1907329859288154, 12007143096050851840,

    26377471328340867789078, 27704323374171087575250082,

    16698741537008806443122649024, 6495264795094238771307110966606,

    1767559851278753740684204294616522, 356793819251957926160193681072539900,

    55825070847432859050184034440524400640,

    7005063276906293497364948035277706626514,

    724295349480722607914989529838465451491094,

    63074718314070991996804319734154570572711582,

    4710492502896520681985837142990727442092852736,

    306262280348546009888225979552203875066987408910,

    17557850797572448170774650557065355110407824998400,

    897284767074804879541932447503681860046708855848960,

    41262148496424910885238533360932130462385764844779520,

    1721420906825022407691021582533801318990940905264157440,

    65621885017296073305936415210180612889995007767343942954,

    2300294582783582441878809203619696625646570936059692676480,

    74564105391671852831723906175451893853322572397711983627852,

    2246282983913595423301542626700582392669279470142155479598158

                                                                         n
      Theorem Number , 502,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[12, 10](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

211584, 158803613696, 4470003680231262, 25087450769107063220,

    51011026746856880905934, 50621569363083174228757440,

    29199419581637037423885609406, 10962576015648521207165916881498,

    2897074457590322299986725442416640, 570469144016296600764916959483674624,

    87372328339498827312145258673398525440,

    10761332843278635994430213513372028224448,

    1094527257116273847074026200323868393746418,

    93928628471341411145251969191805878780829878,

    6922838582274971300562887376127451840124226174,

    444762294191698867199978356557395675489429094400,

    25222201150556323579148483573200214307028208321956,

    1276188894095166511415975526512590077082435242096384,

    58150403780944914591803016961270318192320011679825920,

    2405492889896027043576739666690328464093668209381920006,

    90979946768534826703473491897367681831129975223785636046,

    3165877113462459544886830932071963470167869746240465765002,

    101920630803045581718848914162657089823673167090039341368382,

    3050736654058266641312311721170693674994743893336742177077018

                                                                         n
      Theorem Number , 503,  Let , c[12, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[12, 10](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3791682, 1209085837402, 23639424490467758, 106882054514632889920,

    187486417126728787589858, 166787771996286789659686022,

    88346656964850530193599560064, 30960127600416394096423668424266,

    7726637009853314734245642983042816, 1449360731485252118003180058099425638,

    212878565047574269915601882012169904238,

    25276968615611341928579253465205854300160,

    2489020365757022525686631080521941322176734,

    207516482607365255915256327495571721461336710,

    14902105843923074212137926641524925428400836608,

    935101349022389676082704603681705197097691493258,

    51901859533233195065642333481305660694736373282882,

    2574898966441699734407093983545559208165717001368332,

    115217080172497712307701358668114929117930944226918400,

    4686798350572253787704968073898095177722129074289172586,

    174519839854761339837205462421860474449500346596470337438,

    5985217592228149140081708509275232328880830440342861987840,

    190083559921075547881470295252330551292807908133319531823104,

    5617600279781493180481414011206311652406921056519682382352640

                                                                         n
      Theorem Number , 504,  Let , c[12, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 11](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

41, 8376784, 51250343360, 70301801224208, 39234493545023824,

    11769139203234936368, 2221847307856055958576, 291727877447353315394479,

    28519133604802588034599152, 2180082860424560879159740528,

    135180071048839975476378438816, 6994339928946208878816392927248,

    308817801624322818434609454202720, 11847781621171936784157330501985568,

    400874403870346917218483310713611111,

    12111587220746874825584949659548158512,

    330188902726750395395870169144681687136,

    8195484539406106198328951614449756055680,

    186630963138834193413910002708131937959536,

    3925531352604860356392429944367223543863024,

    76712597804170231030823642725066057618398896,

    1400023119673493900642980926435310251681702000,

    23971627112193180813682078007602972083813899744,

    386662656391850868406863808745451743068197621824

                                                                         n
      Theorem Number , 505,  Let , c[12, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 11](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

368, 33291184, 154538954080, 181794882462928, 91635271555982832,

    25526059645699900960, 4551687606028817499456, 570900586353553630933808,

    53740911951418376297425152, 3979064242273865436579273680,

    240053430679506378621168181184, 12127087185771106822347916662944,

    524264784604149593710007809391936, 19738916949976971907582428382946560,

    656691130537494169239554604006444768,

    19539647570985817631129636767810078592,

    525330810643543579366574347152345843360,

    12873837365577305990745577218318405518640,

    289747963720630046796836290770743277193536,

    6028690323375625639905716686600076565054960,

    116631863553753531449140436456348057359073936,

    2108681757103473505266049446616737960359273536,

    35790280586564051284569729918813760737547205664,

    572571885266855316836569208932951968325138933232

                                                                         n
      Theorem Number , 506,  Let , c[12, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2656, 125442000, 453658947552, 462220954671312, 211480926826501808,

    54866191487933596000, 9258328845225435752000, 1110738471591349924010784,

    100775276367699592904504480, 7232346707439657370657776832,

    424750734399246604013002263616, 20959865998637849981574320634832,

    887516285376674914359212037040560, 32803083883545900890836424410855600,

    1073311732198795667863413728486857008,

    31458259618499290631969563062264609296,

    834224399856173226633752924580270785040,

    20187742802072302139580504567641066629744,

    449119127716955950478044181418236547805152,

    9244914941410348506606083590549206090819136,

    177079440591213743473985149807751865673471648,

    3171956532817322084559414939495153062352648608,

    53371415697494763499720271236398080650519318512,

    846907260688460424083854707630157206771664665664

                                                                         n
      Theorem Number , 507,  Let , c[12, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 11](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

88912, 1553532976, 3634623276704, 2850285615045824, 1088926080945388192,

    247023510593537514000, 37525631077185531644928, 4134135239128434760443616,

    349372831791503826875649952, 23604659287424486586992950720,

    1315826039155571229246222730928, 62031888225113133777378968124624,

    2522539400667778044466695950053408, 89924471148492806069930489571824128,

    2848039691329037155574774539889885728,

    81044656866275774484685020655773695200,

    2092001943222949042791444626510963974512,

    49387987407104123988302814400307489970096,

    1073959406573504929377917961240207489929136,

    21644921427640933554650957429524815464357568,

    406532324429326684996594633005464601821180848,

    7149932560858145793501929056528274196898370496,

    118261428176200331339842373299161813604125425056,

    1846668048465754832304405113404911499541367899392

                                                                         n
      Theorem Number , 508,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 12](13 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

24, 14601670976, 8330760315253440, 385314771232211098032,

    4217589939894233102219712, 17491546015922308382495403744,

    35572019317412311524865201960848, 41623875638054994786793043619036160,

    31201255251996076836665815938246815760,

    16171572444516591115994522506635643344768,

    6132936801180033081464979819122962983877760,

    1777295477218954062142039331879846586908765280,

    407234891173204401447078714630966335423599618944,

    75828943852956727256586863113206467767904090072288,

    11734792428294287874318005191377447998127941870692416,

    1537667828570529280502877759713143985990850345163453392,

    173304825812802666947989657318323657830368305298984874496,

    17026299656950615668471225944039635227529294895031964098816,

    1474952413824985713897963409730165320488243498564421784823456,

    113791157522321741151269820409225139492817630786802444429137280,

    7886633481685613816282797524374487076280676678577276880025622368,

    494828009105568427406060977633600895507578758388081182619681581408,

    28297165071640649884036873041742031534452799049710822414551997407800,

    1483823611276277887291776711122003416889863224499317439411980165358848

                                                                         n
      Theorem Number , 509,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 12](13 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10704, 433358032440, 117455430077273856, 3662609189996989519392,

    31066933092966572983328128, 107331564487695803215332223104,

    189885064531990839374650895829600, 198861744705299647028386239188499296,

    136090830136536495565190472709072561296,

    65337796440753776982705695578277434043088,

    23206926979062041994157109255716381899922560,

    6352932357985338499553729658530126513036149392,

    1384522534584026818992663538744339311979512445440,

    246575500088692326869390549563565819453562170681664,

    36664827090559208660165792104739035609024500051806016,

    4634162908900476988593602324248801817268470289569436648,

    505445473878527532827333669756847458483683147250351744768,

    48189855105483356767464732747746604591327202129891557267120,

    4061037993746377041241345537692343140803683106081534573292672,

    305427535665297871124512935054619693637460294868440350824938432,

    20674593739908650006140038647958003200926386344680812119777328096,

    1268985479275894366265504029711822205596830184930028261795784692256,

    71094293012662467148071388577732221379435321104776611731622939833792,

    3657031349234148779099393749684764507126727393687333504304021276532960

                                                                         n
      Theorem Number , 510,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 12](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

60256, 1271984888928, 276709501560284184, 7640264643950165968272,

    59806691941661877047107536, 194942696685106035131892751296,

    329795213222222890948958568211200, 333239772099526624713483641282742768,

    221420211599992324737147821098107009216,

    103692120164583075994745516868562341670080,

    36051463536425925938428462476103477702204848,

    9687284747305598613964980842632390667938627920,

    2076872464956078729502472886289055130956817655128,

    364522211927201855188063974515573263915254309851712,

    53497805895118009036675409844511261157586536849932416,

    6682113501722208538357815163955012138404756498412550016,

    720999681251550881089895594871691710882215968288294749984,

    68066114979152157797108042372909132037037256458090135929312,

    5684210035676965873839240962879428626900418257648267987468352,

    423934156397879047962132711462866361337402981418831831164045056,

    28473937566154438619206042376839807577748161021455681901984296704,

    1735083183813511170052187764704289634097515808152144091200611519232,

    96551404405815523688840419171187639641465984593437884813004838052080,

    4935122989196325241465495746070904118887105518991712293180386773407328

                                                                         n
      Theorem Number , 511,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 12](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

306624, 3646545786768, 644330580333465216, 15821997977517456214080,

    114546355175483130851617056, 352709392472254050768643875384,

    571060942501482337594749225715872, 557045525297083238494543384463151960,

    359509397651113798938494726051824176448,

    164272206361380982491257312059590096317184,

    55920080932989082671027865474659432703008432,

    14751999001044290736016251464117019968853701792,

    3111762380893729075070113022717607382093956961248,

    538318163109535580177288423149804435636306791831072,

    77984532898719405523516565684382808197229799999149568,

    9626774183495625718930609653558365507878883981925089424,

    1027668061464677462001446995946573962535265166909188789696,

    96070816144158785098219868819493951886355415603843289825152,

    7950836727594869646177058494292508589522890423201680972362096,

    588057416626219493159471647977696012521274037021094612502555648,

    39192962568038488967326104896698593275395524121346221549402326720,

    2371104846218576781041836607162823442452970065661423160367995447424,

    131058212959182458585164003229529605987996651116637819650955355035856,

    6656753947610602348237407112867020861693202849053956367907895717364720

                                                                         n
      Theorem Number , 512,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[12, 12](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1434648, 10226352238336, 1483666364456801712, 32534538659145750291072,

    218295875244095732126138432, 635755784769677611929979580592,

    985886857614970125027631931142432, 928898795745214086046553683564704288,

    582531409010964348061224322762237457088,

    259793515582345643531159553730106436850464,

    86608244033486480344227450395870679601768704,

    22435007225379149588864426726621281841631376704,

    4656872651994793633433672136074886845360653497840,

    794143158725886985913700995175346760771023759812736,

    113571751045795485305206774389083197920873545496310080,

    13857178105940774656634659858828609003563585928771118208,

    1463625559562071478557401475385528069606549153251146832320,

    135499962828495868038961107209427463941958910157707509869264,

    11113912094692329430491701454393757103580164533184488564522872,

    815217877920125172186649370940670995560555845844052703692920448,

    53916294861095575080562049237958092492649840474470868868908781920,

    3238539794686813638707061552640929689008323390416235673625366373936,

    177808601006565098786563684555496364159379090003877975346195677077600,

    8974768022807193010730099921226100623956365360994202881834972165524320

                                                                         n
      Theorem Number , 513,  Let , c[12, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[12, 12](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

99333792, 200261824084832, 17012252120758575312, 271681965292256486111208,

    1468017064295585054450227904, 3642511363292699267474121564384,

    4985519358345266423906748467026992, 4246339920239608886023651412341251264,

    2448809017644084985803567872240666107776,

    1017142690924991376938493165901832682857496,

    318918932984384622687860218463783156292540928,

    78300702016928896746655565661147958257494116608,

    15500674623943154121669939290219653473971594207424,

    2533864416059157804411025234373796639456951317069904,

    348837351204206907198481210057364782463165821288238832,

    41119304390399444752072055341631314506908639910281810880,

    4208624085345896369777321764414075599264621956041234671808,

    378551577749950190913764399778084861885167137201633512118224,

    30235349200599711619658995943296611245386095122674825921147728,

    2163946934956298645426997083462716919663152339666034344013973632,

    139887084113982201361409858933077488600698926923251809914757514112,

    8225546910194100376036442542279315319691665100438292409819677526208,

    442717116356373954721539213185010165040993909519777253030675082604800,

    21932709000904721239930907422249342776290297287337812646312656165495296

                                                                         n
      Theorem Number , 514,  Let , c[12, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

73, 1307566, 2084623577, 1052272678918, 258668122660538, 38268018831162568,

    3855255303983036285, 286446882586171537734, 16587352966624210795564,

    779244299306637609018602, 30611288392669235095737104,

    1029487185172497904072722722, 30201418621497634997908998972,

    784704170928086828488498713284, 18285574743167416315515433005689,

    386188100029179477083280838501376, 7458361483650613444245399968778265,

    132725135738321607000507732287775714, 2190734151449761943846235762249739996,

    33732342899071691997056460153154072684,

    486981660021298290748365077172799454495,

    6620979101997044176881488207852198430410,

    85112769578242885962119399443593460227917,

    1038165986479767204264212824636338883873336

                                                                         n
      Theorem Number , 515,  Let , c[12, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[12, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5892, 30022668, 27950841389, 10221108857202, 2012182615880767,

    252289868278845606, 22325012159175548525, 1492918969852394575338,

    79181110251322879873768, 3451853403326330635870800,

    127109577287484607573997918, 4039280204366619582318731918,

    112693340916996529699520369687, 2799413273484330401554572085468,

    62644042735734381437784743995695, 1275268433725301871727045396776892,

    23815634908384227192398094026481544, 410942859381067114367342776735209708,

    6592727972844081940291799388514998378,

    98873129374210695941043847064796954008,

    1392844096091964063882170421720326757129,

    18508952110893766255676519982114480764572,

    232894407423509728455858578394797346648376,

    2784231886455359694585282400251409807606070

                                                                         n
      Theorem Number , 516,  Let , c[12, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



            Then , 1/19 c[12, 15](19 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

793171940, 900962059717637478, 5697065012312370812670567,

    2933867340992273872469228274323, 314859493089688909967225434086536151,

    11206717459592831795065424414169651030765,

    173488107421982902334026370829027294049018342,

    1391346792498056774170118102940707199693250167519,

    6522368542666912849062739495600111724043497506590288,

    19507750396689962729854640910357592655710801648190134052,

    39762949624747402049104526452059584297660891315022443767812,

    58134454582664366760798037907991684459201453701429448498470085,

    63492291086899824343116842211133077697815224936389963328347826772,

    53535581164192438745653953854182891128518767472004140821301229173142,

    35808143118579707905065133251108170339596739684202095814172256189542872,

    19435074238824956357757005873762435174248460272769187280427924638187173695,
    872541654902165569900558305166910233651628299937565159487818982840002755\

    9790, 3293853967333524297383136295217056948773299982719734878837482214435\

    796453080570, 10604446453911178114041574049824102707887254917857700872898\

    32920589566897520698367, 294772450746120728719212690526679373838134789705\

    175591127554861263272901926098152696, 71514039604103149391429799766849637\

    933685922902737886144783791049820222613396480707995, 15287655609380045594\

    260185015218423330621029371257435722919918498342059752118755997617454, 29\
    040981151275768331770022690421640806676702867933054731512000268887955676\

    91282086763922547, 493951191900710365166057074702484152874141418930431518\
    901143637550494839610414469048195877440

                                                                         n
      Theorem Number , 517,  Let , c[12, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 12
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[12, 17](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6338, 1030390271, 10468425439405, 26540518204442693, 27788253183461900249,

    15495236046118258318439, 5349108092528356744898071,

    1260985888736812287956840340, 217338160113402715773476788602,

    28787833623232520362544578004154, 3043161346799122539573958054294567,

    264400946862326915820685036326509314,

    19329386035747665877197065932101735884,

    1211942141120566472813226726007746941385,

    66207700326638086277964885168609636333870,

    3193319527021436349901508403364321338910594,

    137516835416121841547842312382549983858451794,

    5338569525053236761011895341791781060266715949,

    188391566606920705857088926729629915033505795202,

    6087167251701692922284878853981307584904503920170,

    181239491795863059121479090066456514948477923659031,

    5000578709439115071553032351368913833947886705460746,

    128496967949708559746570913558189700163916813168683592,

    3088984303506323068960617798875490537749832035295335341

                                                                        n
      Theorem Number , 518,  Let , c[13, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/7 c[13, 1](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

84, 436780, 199349572, 31200151423, 2561158451292, 134750442152760,

    5091802213577068, 148512772764543320, 3511807220951358056,

    69739755151539745372, 1194119140054465668526, 17990495916561695690744,

    242341563718758399698628, 2956760616110435485714160,

    33022860741796523062505936, 340611680620161713695290888,

    3268779202291336793135529064, 29373178737760847490241330506,

    248502088529598337348338485264, 1988769865227631547649814170056,

    15118830685383843187775795170256, 109577167501788872732453310895912,

    759623854235199539350134553110516, 5051370389647287350176311185754456

                                                                        n
      Theorem Number , 519,  Let , c[13, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 2](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3, 11297, 3112353, 304915389, 16312275649, 578545196911, 15151247640367,

    313397466882282, 5358288048322089, 78221741490106715, 998786736049468800,

    11363244974048757323, 116882767519366115002, 1099800350129865569452,

    9557991977157810432902, 77335555562406893410569, 586508125413250415040000,

    4193146905508820540385525, 28399974161547975642098721,

    183005732750228802489930463, 1126169882721381623647906651,

    6639951642391613729587389046, 37619577999532885405217472010,

    205342562925851297636123889280

                                                                        n
      Theorem Number , 520,  Let , c[13, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 2](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

145, 127099, 21342592, 1588909079, 70909889147, 2205320636989, 52208032174122,

    996105810421248, 15934196727973843, 219922126690819670, 2676156436311330659,

    29197535724600008576, 289445158202213980379, 2635530543387168915503,

    22239343764937319783552, 175211739485115758622027,

    1296967235363163721733420, 9069013661768534769650694,

    60183396432227951980589622, 380572372842204987931275520,

    2301354004443033795750367872, 13349835974792700506178039936,

    74494292374597470537574979363, 400869801287440633953678012649

                                                                        n
      Theorem Number , 521,  Let , c[13, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 2](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

706, 387603, 53229543, 3516700279, 144647851784, 4235126929757, 95655938404998,

    1757035819425070, 27233765006521306, 365960562683005416,

    4351782196094946534, 46532386087785681629, 453153940415558551936,

    4061168323867753041982, 33783232182057889557559, 262737661211164631796842,

    1922052058782094217248537, 13295381922542619540284672,

    87356070357564978660224899, 547335596626833522154068256,

    3281599093315419592172167296, 18884988462375502450327307776,

    104599099136158258475887592704, 558951777096727516431131464320

                                                                        n
      Theorem Number , 522,  Let , c[13, 3](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 3
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 3](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

11622, 30538304, 14385490735, 2722733198560, 285200520515212, 19535217803943264,

    967285573836507408, 36981972085498410816, 1143293040607563653218,

    29561934473433074344688, 655922947190492268802856,

    12741158873266989295764832, 220161056380042431013406692,

    3428418010888022525876992064, 48633636921577206323953235274,

    634135051695320232165263371776, 7658552993954352987848689212763,

    86233231191728114653496832080160, 910382579967044842921958970903572,

    9056023917303717048084620522619232, 85250545071302792634059112275075716,

    762369789133796203755703267979254752, 6498623931057389166721671965028379070,

    52964583977612688434981340115427131792

                                                                        n
      Theorem Number , 523,  Let , c[13, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 4](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1152, 1048849, 220968832, 21331021824, 1238778624384, 49899294124288,

    1519024212898337, 36978548661424512, 748955613195929216,

    12992945903238631680, 197377475981175100800, 2671292529003902338925,

    32656524080854113304704, 364688539804221739537536,

    3755044245875321650207488, 35928159247492345660442112,

    321555238569600098560754686, 2707328476134645173782376832,

    21548794414078787689113485184, 162840017006099554815291217536,

    1172705520231734315660509271680, 8075154302390689700687287319808,

    53325054157035408438736139359104, 338593775339114327552271637570048

                                                                        n
      Theorem Number , 524,  Let , c[13, 5](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 5
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 5](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

57408, 8918451296, 69520483413376, 127462460575405295, 94625183065888427200,

    37209473446001710266624, 9064577976831471951287744,

    1513310424892395885378008992, 185624271072021388730804824704,

    17594776315916278226932679201920, 1338650824457053878339552919231744,

    84191814064177703727647212748319360, 4480662321091365657432504368914336128,

    205638959653153954370625894639541192064,

    8266647502915738500215487476126477261006,

    294896143653645253746104842006542884897184,

    9438565127746104915790103178650247021083200,

    273606647042125830965211050533350815252169312,

    7241969145251598273557687989231955403050791040,

    176263783067333823069010161265681673490399664256,

    3969482901478478550072420641746296163747396310144,

    83165056431939920207582859420886990638005309925248,

    1628880173469704051166552819645284637088744348543488,

    29954192220378458274650215598518896017030705874603360

                                                                        n
      Theorem Number , 525,  Let , c[13, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 6](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

209, 6161728, 11297688876, 5880107903204, 1416919755694665, 199855064432589815,

    18876071622313863425, 1300779357884670318524, 69362614673735529384079,

    2986025695152963523853504, 107131508323554728313172453,

    3283007195956792938872378112, 87622446988710499846007815380,

    2069122236395414389045940816106, 43793293986184190564717895177472,

    839791589686772178426846126523904, 14724525199509484140154521879212743,

    237903913875244721043007726732084320, 3565910174340368309485775150614661528,

    49875212379958757651671301082725546123,

    654285179226115361939093368535802897600,

    8086812672643077574288020982495493174797,

    94548534484744975813667835909706048689191,

    1049424146640310618721425106582327456548343

                                                                        n
      Theorem Number , 526,  Let , c[13, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 6](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6555, 63053528, 74514332859, 30090088783951, 6110631003787008,

    760172017867333801, 65136177010786346052, 4150136471588770503285,

    207388071491345730320811, 8450622373847969121874076,

    289181981433912469276706688, 8503481800892020050215941309,

    218831474600869641192768657789, 5002262307013920053176055072601,

    102825133564658485696767145987968, 1920321275195133486012069561264595,

    32868128694717004720872355957841764, 519449984315194250274417860241511872,

    7629244754367104479293387353669877504,

    104719960689385031559456287792545635033,

    1349988742530147805052189504434531962588,

    16416351334678190412709962746923046434048,

    189039006958013086024848790731471210375921,

    2068511342260438406655860182741685775020416

                                                                        n
      Theorem Number , 527,  Let , c[13, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

30243, 189777244, 185244874639, 66652215144208, 12500467605033287,

    1465668705469267937, 119900924597341417920, 7357797668456062867205,

    356359787922645000150272, 14140200772039070566264659,

    472905250411426416566523349, 13629489831368334953319341897,

    344568816613738991017440191844, 7752462119563964289810313712948,

    157096143813250043116260784650095, 2896088962321746100027279868670441,

    48986715192691943905637787189453056, 765838961943520714700750353652416234,

    11136143557688499007405261264120008767,

    151448646371809965908116662519309291275,

    1935679447164668886476089919329599538217,

    23350690997170833148931439722601482709504,

    266882079470405065262729110502437538681181,

    2899829666140155627740453221241255966324024

                                                                        n
      Theorem Number , 528,  Let , c[13, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 7](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4, 56157, 45245756, 11249902864, 1390001614024, 106283498023056,

    5689201518066616, 230417636338903952, 7443678300846924048,

    199208163554323438432, 4543056129360100787280, 90233356776274772357589,

    1587950261591378524406140, 25106515231464677467461200,

    360715790933496171329203104, 4754409234014961802914522208,

    57950864087298367804582621852, 657693294642007852363205941392,

    6991073223487249906548029076424, 69958833334906719841020160916688,

    662009433673262042912484248502660, 5947340382368233494687784274945144,

    50902440759176053256359864871554512, 416358472504328470317368131390709280

                                                                        n
      Theorem Number , 529,  Let , c[13, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 7](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

340, 991848, 456501612, 82726152968, 8275658471784, 542349520254160,

    25769441118218760, 948358809632768024, 28303666414103800204,

    708400703364543280552, 15251138594440563032976, 288068334462664670720536,

    4849544250723518275037416, 73701992730000481702287312,

    1021936102613827276044156388, 13043123420102760763001925712,

    154388096190116407530512526884, 1705742801688391293168630047264,

    17688678570196326100537270816052, 173007024182174448348920955822360,

    1602751003073887424524810359541256, 14116756691470564566289077143039928,

    118610078251704548308107479345483544, 953505154890522600923500503319522136

                                                                        n
      Theorem Number , 530,  Let , c[13, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 7](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2152, 3745988, 1371168456, 216421787472, 19682529725736, 1201527247945780,

    54003018557452624, 1899754571526903196, 54607157236583964072,

    1323769115482868152468, 27723417011323363560976, 511141995419802007326072,

    8422702516086902114190496, 125580481964217460269345324,

    1711521293056165664756881440, 21505553135133996084229425604,

    250949378395597597964628093392, 2736528124784975846502058681104,

    28037521234656179356190245572064, 271177954135406088887742217960956,

    2486254861549616924143529924206776, 21687388877828114903886335972847464,

    180575009815447679679280099091473360, 1439347729792516369979382209811502164

                                                                        n
      Theorem Number , 531,  Let , c[13, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 7](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

450840, 98373539760, 1195329532711099, 3383490136748601840,

    3793738315525732062264, 2204338295783417397744280,

    777964224116438938525855200, 184909848300671614654917454856,

    31799469697684872655009164053040, 4168896512279826159125829208781664,

    433411434041146636783409319690691880,

    36845496492208754362509921113993686120,

    2624720710741570801061796081793424868960,

    159811769974670735512254976539026087898780,

    8454101703474955234179659918590227502906200,

    393917063946924834733543832789845709864861520,

    16355191656010899881036535281753247355012059640,

    611123190727137268531585312972201421017451884088,

    20727447149394235442332946786414486643190530614280,

    642906502158596956434217120758501980048438558585280,

    18355998938933231684743619904304228740781945345630600,

    485229543431351825839786968703933171447138923025910880,

    11936745170715655257170813941591996368515522506707815600,

    274527765243041224419991054077733452269628776068929524960

                                                                        n
      Theorem Number , 532,  Let , c[13, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 8](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

280, 14436375, 41687787056, 32213884708440, 11078979968421680,

    2166657850893700256, 277420422428805326648, 25450234296339303938480,

    1779584539276402204510691, 99180171904151388529199560,

    4555948235053293659565818008, 177032772001732744242712425112,

    5940033966799342012897767797248, 174991729608335619106879325488640,

    4588718761595625136816405208282120, 108337667276150194901103499972930937,

    2325312712218230800528164288032977624,

    45749386766464342243255894375586440168,

    830971421123287266754035389054393123080,

    14020893364254219185905046482637133538920,

    220959685898590489473150344903800930865056,

    3267979517964177013936629700610597070611096,

    45553502990737903031470822369066406391162608,

    600743486405959163628383770450773420523006280

                                                                        n
      Theorem Number , 533,  Let , c[13, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 9](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3093, 7857564, 3978328420, 839080234560, 99480969393316, 7768534072053290,

    439955625038480600, 19260314823562571880, 681745942673316954020,

    20168063641657891746244, 511386971964620267108154,

    11336201257744499388531992, 223202498479599930303622120,

    3954216181490402188503196640, 63710008828024686703599569088,

    942004173263454444507377182740, 12880153802087081285546587488916,

    163933396536315899761130773243720, 1953284410473876421581595599311100,

    21896522444641493156633595611844392, 231950880913031545002785778129029017,

    2330823857258466945143925575820001556,

    22295242446052387938076910042493455300,

    203630928561886211958887403691494191080

                                                                         n
      Theorem Number , 534,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 10](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

86215, 1419714560, 3129747365888, 2317918951636992, 838396090046308352,

    180482665304875614208, 26072069727582143815680, 2736467509108861490840033,

    220686822298892266264961024, 14250100160197480006975832064,

    760223090622846482962939142144, 34341285535900540599228130304000,

    1339644147061472203060675593510912, 45859655031942134332735068198051840,

    1396102459873965325956445534811078226,

    38220880411351080587272231327279251456,

    949955701897834682992067335122891816960,

    21610446060121029254804916891030207062016,

    453154542966854878524915488467298727813120,

    8813054329010521829317447334112115105865728,

    159829228795175736925523559775034320858161152,

    2715919789757090048049808959000945877173833871,

    43427122492355607395812003481579861601920933888,

    655912488467353736993896325074906263610447724544

                                                                         n
      Theorem Number , 535,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 10](11 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

26, 903454720, 96384817178234, 1182389333268106730, 4151851443162943803738,

    6257942090170501340396150, 5061784052117481635767212480,

    2524559566708990914396147181696, 852500820857300492856793559506944,

    208330229325322709019895576568081846,

    38710720322551991811307850327682588672,

    5680704203452528574690065774554862428224,

    678315463057296170892044683863993614191040,

    67506911702932916177004057870670860734076672,

    5710700737166793810257997105205206151168628102,

    417398856297406923541570388684031258661194006528,

    26724439178665787683870186591124950283993912156800,

    1516523272409111540851921607347555835132603226789996,

    77045598084517429275725866460150825484551069027366854,

    3535087781126884213193185289646107719393245947616179840,

    147612869665028665141902946034714339757006933622215693926,

    5647300065301532246571062821592379755225125823317219512896,

    199130465802090529522286948236010288623329575935465606938624,

    6506089760992467703653624075200915755341858157389936889311552

                                                                         n
      Theorem Number , 536,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 10](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10166, 29047943296, 1475039332859904, 12134482733075352000,

    32842491496990757514880, 41047098298931396458537920,

    28769646655174863788712173206, 12799453637167945214153856944294,

    3934580783692836208886594655703360, 888428838101614298335919885788867962,

    154269686200886012744462700975631646336,

    21343533375419089773010833829672771004890,

    2419773423439264664068247092549456302249802,

    229966321445652725694040826938713247768657920,

    18665570918155261185102636374045949952040790144,

    1314221442476701749119031961598197022365015813274,

    81331461560426051602844028166470285574470705389568,

    4473932090836142215196375328415554745832988502533824,

    220885029376727933818115294128065515019504623883796864,

    9870640273557440900260188994790258310577626408310291072,

    402187973952620429525104020965006756757060597542348426186,

    15039793503949491778135007148151785616816079995347842562438,

    519146205487046541207102130590514620145680355382751954908288,

    16626787841740244975013285211588370444965103988653661261373440

                                                                         n
      Theorem Number , 537,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 10](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

267904, 252800416960, 8450219621438858, 54757243390672433792,

    126304587147933788684246, 140437068735658714895884288,

    89921462563855462691614442304, 37201025175372696198586402186298,

    10768547347198311713350544247957270, 2311082643953261755838840578415433344,

    384145748272573186799267112362958423818,

    51159878695711658341840639161010343085574,

    5608307300279422982292921030798618703732736,

    517256053153426806380283738386834109469522918,

    40868295563535153363942990979222826479956962250,

    2808195033429889108914451680351314400849815944890,

    169970831764255608450176245802804412048040103736896,

    9161633128974790089351302027200144067208902629896230,

    443935569511300235176175239545213440080284441779624576,

    19497669773192055902159707384218702047515129634687115264,

    781789625582999548815778033633311272398374768663523213312,

    28800756388389218836182472460548615866406527778167995269120,

    980343460827099864751643563731056640019775370650339796245110,

    30988757114814633191341188056486174785397018067656659248484154

                                                                         n
      Theorem Number , 538,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 10](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1203904, 717813899862, 19827664285227994, 114852605194011754496,

    245468736752064797264000, 257996320584956415090424640,

    158116029312274160535641484966, 63141140509714184308047880339584,

    17748987184262121068003943247414010, 3715644179887522097803687576384332886,

    604517264844076616449497395078203883520,

    79015349409524470237007229518058289654474,

    8519786076069626072399166324398013585006208,

    774269033833192256427793883623022414843936768,

    60367913368609627644295239420227333719465462298,

    4098477277398259797946930647115159654081544499776,

    245361915767912764608295833435882662181091471035520,

    13093036745662800467208721785772610345022153539125248,

    628589061191347922646388465801585009346471802181377484,

    27372238266044015728888232264859282649612291168179279094,

    1088838666395269056106310869134453782932455970080757948410,

    39816089195399105073811184914128644863898761422021077505962,

    1345933043475738125690181310077674891646224553480836629078016,

    42269527972859226393654815013948868654217009181481402124691466

                                                                         n
      Theorem Number , 539,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 10](11 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5041738, 1991657414656, 45948626772700288, 238963829183056853414,

    474304747046944766477130, 471873743787365026117380864,

    277051574259201278127437090666, 106860155766920196465549543389376,

    29183416838508630939013225216942080, 5961432892799422848298579556759673536,

    949593283416642655247288200547129303446,

    121843982246136058731261671748097219380352,

    12924477821631197157297562903727567980945718,

    1157521470839591624042046325112762802130488774,

    89069659729696448319768863074932789838499558720,

    5975397852670474306374257848045811031210965901312,

    353856777289418896195236632982682728515148635327050,

    18695144876071655991860467896414447554209957678125696,

    889331582037297190651292183656262277330905555349542422,

    38398427983500343165621354192763545790306249621745279680,

    1515427765727641573706862642396672598186362609779281084416,

    55008704315550005796704516189340010378277355158366551696884,

    1846735363008524363065061054636393532507181441630064101832022,

    57623910258027861036310506822813202729145469639740697648222054

                                                                         n
      Theorem Number , 540,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 10](13 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1408, 73487213658, 16777254395192918, 417427109635465256960,

    2800568917305665501936822, 7673130855490950772986986496,

    10830306923757492104284032799018, 9111847755321648372835075902117690,

    5044289370830280871637259624950492992,

    1972110327344136586567333183359194424576,

    573959237971558298447114732500302288930880,

    129493411009269834989624487522464987842294656,

    23383835188191480905437232643927617785649733632,

    3467942311644681367493556701405135091915153038272,

    431407862871903398917153439873206130186470467054230,

    45813888856470911726737260655889632332415077699793152,

    4215341944192995406112962470911053312623524717370445046,

    340312901036832671136626479278082640332897428689679437542,

    24369870790175492861385169005079598040805523504603688304640,

    1562622230237982153362743969307717604429153527533131163088634,

    90461645259044681029948571278817641359997567432127443826572810,

    4762569436115735812555883742243633210408833513181890920174416768,

    229499769918482083934247196685808266311822799649059943626011846016,

    10180776740120515791162973776770014483806066090134055375896641550790

                                                                         n
      Theorem Number , 541,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 10](13 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8602, 213908045120, 38879607269207936, 855257930821256981322,

    5295181768605808826489050, 13693197640407126727441761536,

    18491461442407983399584377856000, 15018373771298717826772567363196470,

    8076559570519990566386353198215354778,

    3081519183495893834133772392810819171478,

    878291518129814481859814862053605280661504,

    194586887377090767894957622794295825035018240,

    34580865476837437461797915443957776252271275750,

    5056105875336555182316679717577224718716971147264,

    621002420729780278109326654200520878921766940278656,

    65192429148437824771768040850896852333081673792387338,

    5935815868778530926003475357646288135701079676226437120,

    474637161147354420246138737696411876144245259261336720640,

    33690537011491550447071002619647314884837413510940911735814,

    2142756159991083941345810750284711732205816432583147221410934,

    123113109444952608864857026493087182161352786278502501777334272,

    6436197829317912960692074157380232927394412570775888851590629398,

    308120472709990626386016825865030104428962822066517782730953809036,

    13584698569109817880532286536113890224960152684965749498670894740864

                                                                         n
      Theorem Number , 542,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 10](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

226688, 1685248581632, 201581392406154688, 3513105067291721680086,

    18645376291938188604931456, 43116401975950330264204817162,

    53427118892835079029886667979648, 40505095899529150749191114368939072,

    20580474067650581854685047637765750784,

    7485138430272111098615220565559421569408,

    2047500589063993177288516770618770717288294,

    437678198822130374629470855558090400320365546,

    75366635155896994732112114909558514478730395840,

    10714373991894162481219360452933878585531520284470,

    1283211999730786688413164436986393398958356576514612,

    131675133977053197806122897608423833777519473747422592,

    11742965518060108531689496433880958433532544788361163142,

    921325025411381509013032273882999354789001205452949033270,

    64264832821636786459824887644122969543084429510143843424640,

    4021878520702956071285022577370400479715662957653659235083136,

    227644865561795769851259406414419216291411530042818035149631482,

    11736224991778915471876646574225959002175317738458817762541644032,

    554579443665577214022906041162500447496998806049449725307986698240,

    24154273114174021409270836980377619826870714548114074963343433138176

                                                                         n
      Theorem Number , 543,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 10](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1018688, 4576156970029, 451444048486473728, 7046284502176275947456,

    34732160099095797003378240, 76087391400185391508289143488,

    90420131802778627778541921329472, 66284293407024096965581497849569472,

    32755224888313223840337411815731421184,

    11636394032372861906433416787477803802624,

    3119358579799590738163014394525211347931520,

    655150720935778063131210209219480504867946496,

    111073676010317939458051993059987403163596699712,

    15573294624209183603806048449835696005487421534400,

    1842062957736172260709526536296636396278624889600011,

    186902717164923636307636018108363128093426989440135808,

    16498028269624047301827444709723517117128186613966020608,

    1282285032538773639290422235874107583080768362120930148352,

    88672208984800289002111747323992559263743679563366552856128,

    5505152874685934210783835755939236408366187671637638906340544,

    309295960125076236907756272056452477463702081886409943588577280,

    15835863889563729332948403501030548617917346204927159911684257216,

    743482560732649388271172369540677489954849956643421242852480485952,

    32186276669323813671351559976504628615661793811966946257682830311424

                                                                         n
      Theorem Number , 544,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 10](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4266086, 12177717103488, 1000483281996090310, 14038806776390385131520,

    64392598826268340530851712, 133790486341155058914773564202,

    152593187814612836558149496031756, 108218907323745044392564648513927242,

    52030855898289187759519971035239928000,

    18059912856123845192547628625107729311830,

    4745490792544127138863240872214215826235392,

    979441244556577528035577659710799294110413578,

    163514356966111144427795840241734983775561039258,

    22612986997332589578659388654028804086456387209600,

    2641903248916824233717464249305438439946396398273920,

    265075711496209024603609036766973767307255908381368320,

    23161124686652628693674658070265546857364246373074774618,

    1783430830687529211929106957110739126875520075585917345792,

    122271393370422417899891543311221442127195907011968948786880,

    7531012128009922559571363054846327142290912585630531747116928,

    420002735767277377795143401308909335000444919325182896102716442,

    21356620097926306656212661507330187751622357625290471291479205696,

    996252603376135615230173973609782612887919116711630860657579885440,

    42869848346094123380204804930540693944396909445927625009764569066880

                                                                         n
      Theorem Number , 545,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[13, 10](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

62799430, 81556383766198, 4769594235999165696, 54664584181888621738774,

    218304578956501581999590080, 409350894950763490826197999616,

    430978167500595300027129600081920, 286497045634848465940859401141841280,

    130535888323826626168391516210149854592,

    43289128127140633981557463905470290303178,

    10936096618303320671559476303154095983877146,

    2180851403921340962742103082563907708518089750,

    353183647636267396842867251963411065021010313216,

    47535510127537803539047134323549364419135927502720,

    5419670156016925425066429467198215954179602696724480,

    531883051777294396085405624909033469447014601845780948,

    45545304204536413833484942977933557385179041761053031494,

    3442763688738162871588549311737370660769256888169438602624,

    232046869544900334045835084544576167980386119616257902302912,

    14068820481472514978857395179036313453432011067322328759623680,

    773210025927223424049444132210073116048103426103066858376696250,

    38783851074787406608504797090689423638185130825253794222510768714,

    1786265246879383936302583848651433236506168240958052982990548144090,

    75950630820744750205579358542954430177085188838901677886605787555200

                                                                         n
      Theorem Number , 546,  Let , c[13, 10](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 10
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[13, 10](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

223783286, 205683565635968, 10267639235679144000, 106873112202097821194362,

    399277783204693483637783808, 712378123508795892336113327910,

    721346848417715801648056088813568, 464592782175454634585971747827849600,

    206176554766940693812355577404341340406,

    66859141808059167123621501899895475861478,

    16566865472744219759571531204407075916195712,

    3248230096639709926742715848192244725835588084,

    518213708002886528573041924361494514861280111680,

    68818929012668197510098793063936395486090596868096,

    7752151109132514690989563253784737287638302225296810,

    752511338646943776704939540016837311587689316064997434,

    63797589955210537428809060460940190449369943404805365120,

    4778484670639758054790899310578167485190491081268416510912,

    319368811375328990670728968172642333556154756668042851067264,

    19212370153700977038274705879686954816835958877541686393559936,

    1048256380917793800887406853648563053904402419883081474190057472,

    52225288967261140142124529649094377357252124514178553697444959782,

    2390153445300625249363668046776860491331341906801204921448432303168,

    101025602864187861322812486939097664315586298586786694629918187255680

                                                                         n
      Theorem Number , 547,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 12](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5, 190164, 350739740, 180310506272, 42982841126940, 6024240966688881,

    567871121586153924, 39206396572491620108, 2101317778821832094012,

    91167028887859698770944, 3303783285870785225303701,

    102454489327608398399884244, 2771570996119630655840086268,

    66424788244851249427662180340, 1428501970762905217580034058464,

    27861110484740622827331641991034, 497263800403839996617525420426948,

    8184285706715804610381437302064704, 125042028497938111411874102019039232,

    1783666379642558562696170040973605708,

    23875106493697629805374056826496657930,

    301220597586364936329837159241175087892,

    3596234281554745429472518210422669591552,

    40772480935169335045099065904624538940396

                                                                         n
      Theorem Number , 548,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 12](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

650, 4750382, 4779033210, 1736828942774, 327866370169000, 38716671759072454,

    3195269342626531072, 198182008893531588192, 9718498922270345969554,

    391058070691155261428934, 13280971411562746365415424,

    389152735241270045090205160, 10012560205179208967750100992,

    229466892980940388951157866496, 4740094096339959516904051816834,

    89137228556832138159487379603874, 1538854090553127255340373443112842,

    24566395595346242943610123574407470, 364928777829374681654780475733373600,

    5071866724594353395100482759892139776,

    66267951162822872545607497703641468438,

    817443058699247915794992966324122904506,

    9555836441328555987422646597394943687582,

    106219201470207969741636153322234514248802

                                                                         n
      Theorem Number , 549,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 12](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

5018, 21162440, 16594936862, 5181277357230, 880174555211666, 96045960093344768,

    7449560108230256640, 439286489745837007822, 20651934082658634633240,

    801664487813124109210816, 26391714568612931227713822,

    752498383255420790108186880, 18898264405308265950934666284,

    423831646457810581262934939810, 8585772508977808722744220323798,

    158617182590892760129830494228760, 2694345585765359278264357579840626,

    42377617415720539408034584368438426, 620929291861139807124946828862505728,

    8520767961368273831362959758950027214,

    110021538064700180254703124056146389024,

    1342263996312275653764862496045360523846,

    15529646357423419213327847761369108684800,

    170955462069761555824497505947996870618218

                                                                         n
      Theorem Number , 550,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 12](13 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

250, 69350194720, 36940753882055680, 1745181968190253895162,

    20055810713525111252390626, 88256497300361688058137640960,

    191255307847630767929817469394980, 238818958408130695048056472639425280,

    191060351465780422906878682516587269256,

    105627622567748790296199268818184335667200,

    42689641691564400270437920034692202158807960,

    13169469725089824366408942501799322324454957460,

    3208486160058740453431238614620411950028802391840,

    634480938936788241573705533799745276714408665923038,

    104153252067871873726730050714123391359455077648493642,

    14460079194918203585236667191131784257184577131587742890,

    1724813244933158935074122775097043292809903218287642309380,

    179145293727087559546813994145605498639196437631839062589440,

    16389513220143425627103804653940347980807457293497695585860202,

    1334036119893082128809914120045577783066449659890552703455221760,

    97455967777902135780821954717536543875418553264901674912235949440,

    6439221985536515250311037821260504524411484108402502891729865180040,

    387441575046865126643050894090277603501294693220585055267621485356200,

    21358361095928431713965364416687610300397236072562644140281538492193536

                                                                         n
      Theorem Number , 551,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 12](13 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1930, 218241147150, 90406792964919656, 3737884200873209988290,

    39405572818310922461493940, 163012171714542531254974976850,

    336984410033107412076107229671104, 405289130390166604137323563780979162,

    314401176422787659921151140893893424810,

    169372536329852612190863098034990871986720,

    66951139484938400419921427087012426516240230,

    20259814784235201264815996976606798470315154886,

    4852949799962807487240172196953668530301395722240,

    945325461797737021375771046627202787524067175195670,

    153097934779398775324837509142168690471312723774791680,

    20997507343816597123682371331942197247832261656960962560,

    2476967557632738140629474634656464121846817015855469876440,

    254669261570654327638563684695615574371634143575000372802176,

    23082668406852917664240914544151632366650451765280140290421740,

    1862715883073056836690979498072893586159576008225875100950638050,

    134995380998400634215960467930964344743243042278309753113092579990,

    8853520496271447709950176185488009175590523472810238844616787970816,

    529023422853257821345608431398107957460079917962070226904519396806450,

    28974362453887415412270549231522028218927427481886993889371729707693800

                                                                         n
      Theorem Number , 552,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 12](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

73140, 1992798983550, 521168295897144062, 16750698456738815123456,

    149674128938950017342386600, 549423834908809699069243868640,

    1036286763755907414717060607631010, 1158274205987489725216615249241250816,

    845909874256704332938077841952039374848,

    433116385885568083036652618426669567481850,

    163901127338316435288339677374809044672212780,

    47750490883477534631387819029845838975078689050,

    11061929106933148071179706522177415764821430832934,

    2091653449455326310762063757927969017683104944868890,

    329826830528358064475669852144231437824845176451148770,

    44157489324341519255605572095963059430336599799942518990,

    5095897322583474695349543675609029642492748370338328073120,

    513511265411721871910509374123175528461561987145174026406394,

    45691244258208148354372771868341811491937623213352681656470200,

    3624750655080420938386211027680590292856827468840558894952963890,

    258566248694143022565915167611621103817661306058885794964726296500,

    16709562485262258512573545002999276663440849930013279148726089655040,

    984787255932945328283493888805175576389871088724159619453621919836160,

    53244830971903807850523521120407401749498168970768813696795713479022098

                                                                         n
      Theorem Number , 553,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[13, 12](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1827070, 16531861971900, 2865215426242406400, 72875036882371092998054,

    556822873781490717867549312, 1823113113976907506501558391090,

    3147802194890694080915937423871040, 3277218446680105319754984522673087390,

    2256980982077271711166474378036418585046,

    1099684935505959233059484292813270574057302,

    398771906703989197484541308599321282080567950,

    111935867050923972068440802134955266683207312280,

    25094141435907183463046316943289660373700124116360,

    4608233555507455605957502142919925385209354636929336,

    707819658532868945796052739237014508574199183079774246,

    92537158795669622762874251037676002228049799963008692000,

    10450299194373661546687668323642616382960121997836297579520,

    1032393898929505179186573406450503451527488110787692702965760,

    90198925098446796728126893429386828759112140690833665684640512,

    7035875535397551461846461413914126376426517557280400768694467490,

    494094554379694981349298821538845599263667502729528380224969801400,

    31467890812734124387528393487514019288622318596854258303979567349760,

    1829465566780861937498837444641718578583859553479231426816877448176750,

    97658459381125434394240660551254071225507887787960052053143660022922722

                                                                         n
      Theorem Number , 554,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[13, 12](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

34058546, 126102450065000, 15079383297112161580, 308332495312740042004160,

    2030899335450354357525543674, 5959362634039698409184777982918,

    9448613690517785747542355220925950, 9182733266806780694374637240960253050,

    5972940906701315082049172215911195648000,

    2772712937055015533251817888728329802139162,

    964365285858527633685479622761306809003764918,

    261008424173912394991901313240995072151571858990,

    56658727479465510691991586691439379255916058023980,

    10109879877859922492725459394834896782303309679788190,

    1513229524642088093323847519733606067600604724483382176,

    193252378429689161326718532983709165465556678867190298624,

    21363045441994792873924016511294232981074200614563724016410,

    2069565042394974143067350655985189319703308670866630499843460,

    177584594131990640225235742563969884615249773231165097564986730,

    13623198750157363994685303230679566973949218574928559817402975064,

    941986375031957885424321196149580837444241843859960459743882695974,

    59133484548763088455824412531784276809637704475619092186912155940030,

    3391791429146437136774642940398604371385306225301829885884291847895540,

    178780039933400044641591240900443355012236423435703272112448653104021930

                                                                         n
      Theorem Number , 555,  Let , c[13, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[13, 12](13 n + 11), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

134899900, 338528675081410, 34067844894131300728, 627937278962617003552306,

    3850984694299695756826961920, 10715811724900239548973708458090,

    16299083621430976695397917064784010, 15316424868056893687659313501148577432,

    9687627853876145708575509996727415898976,

    4391465931980610215289364542403254481266600,

    1496347185056473064329372397972164290890566660,

    397782747056323786364250461397943578037578679630,

    84987993643361531177496127356623812153856245166080,

    14951114194186253792635917556803488608966735167111956,

    2209445477183964501470673718270922276630076582137313280,

    278915198581966427266550170402642948321991105694109491200,

    30508510227336110937800076492996601989260025769930002204200,

    2927033512776722626632182162622825711569405541021253235387902,

    248928671943367104049912695257432063604042568355245968854803570,

    18939073875371592318141335929758910030677129794667203924348570400,

    1299537768339046213049529630714057190591773312536540032783550611300,

    80996851673817533336635863575533149623100077335242547689429421706240,

    4614827216855289133176225927978366592637097622749167626570259456469834,

    241722636426897909295778113035402151415141396498130535243685592728091112

                                                                         n
      Theorem Number , 556,  Let , c[13, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 13](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

520, 88170472, 676953049136, 1230612609825768, 916557040101347840,

    365153418785496924856, 90806600075645102393160, 15565269694097292447722472,

    1969067640267109249880358216, 193161129890942068094359634592,

    15251281442937285314782493003568, 997610193966148021662069977072360,

    55315317835098568805572133197468416, 2648701314636421948238478525758345632,

    111218270565150179326354224319100988584,

    4147968116430651100593786356453230337504,

    138904387686788727569016494487881957561392,

    4215430386405746349057498342551678201812616,

    116865790358810879949743987261960919646641552,

    2980415009152640103280764443707526593995072976,

    70349716185269238239344175706020207499555946752,

    1545204562479248424245008238470329510888460921224,

    31734203662140749667905361808854394585387533116744,

    611990625372949562100448115432712014980770487468192

                                                                         n
      Theorem Number , 557,  Let , c[13, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 13](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

28184, 1391389688, 6480081136152, 8787161911774904, 5361509215306116912,

    1843393060294476888808, 408708361020555577439744,

    63849599342389041429155232, 7478022039313171971313456456,

    687138359041927745034030557648, 51277611235231940169822285089080,

    3192664665542868087653075338020560, 169466708206601186796354881801539472,

    7804547204570540726560474862026386408,

    316413494859622375446559963920050984192,

    11431452707993790880110555386321133279856,

    371862136473099414392522681785901889386616,

    10988837492329471841761091401362460092689920,

    297267404120449989225639590060907766742497400,

    7411046285040021360350951289321979383737347488,

    171279922917186925446856459164600686617559292256,

    3688846019723549646104798624142988752824038605752,

    74377998462049944026688095171668096951380603490560,

    1409835711200761224139657002231603871742929711112840

                                                                         n
      Theorem Number , 558,  Let , c[13, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

168168, 5159359768, 19313112527592, 22914132654348832, 12741094102170906576,

    4086347059359115608928, 857803485293478768544760,

    128177739244376542531476224, 14464737561193679893714807200,

    1287740952734788289520767688288, 93502031723595550613759165571048,

    5683556178625075819889983638659144, 295331092556542462249448342939267208,

    13344568658111378692552115101775224880,

    531810345390256580452794968353589968824,

    18916249793783759727879518867685439376640,

    606644308843623824051097364655791681576888,

    17694098142877114528872061002301179061154448,

    472920893719112847473941079022037242027123408,

    11659255550468383767316098604981773924733224776,

    266678473396784245090743867762916031966845611584,

    5688059200683070838254018862686080475621064790672,

    113652647563692624734285576264676852948290993351400,

    2136026898513180279638289370112359789600488416061184

                                                                         n
      Theorem Number , 559,  Let , c[13, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 14](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6709, 38413446, 39275216335, 15567634620786, 3292963000777959,

    440794121132016391, 41438116385155698394, 2932220578438996738471,

    164026496487264421800083, 7521049467593847283451667,

    290608999428712104567858625, 9670373860139273498743215120,

    282004673014460627131138039652, 7310397533644079569450196842631,

    170466427153738017787077776204074, 3611398928146927500626006822582172,

    70102232844567359574630368216545972, 1255949325016298820839353498794231573,

    20899705546565102297030232329729098014,

    324813659686693821847983943201960398327,

    4737677019373673635954188168253317739003,

    65133371093233365771811873926105947197428,

    847255605524342945779799012907484471696573,

    10463812498322786911999899280843726856952817

                                                                         n
      Theorem Number , 560,  Let , c[13, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 15](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

644, 165225749, 1788577406684, 4397908348561027, 4306729983936969180,

    2208873361489746217526, 695566440019703121662876,

    148956738088506854765415900, 23278886638096134566109111868,

    2794165230860283614611013159957, 267704451443036038139546975258664,

    21093638914909860550012859727876574, 1399769121092748077133144953376352356,

    79750820497570285700959497387375092142,

    3963547421551011945142423255294578112420,

    174128505067943729047758439905507725724397,

    6838709332155539460424554825549749896565180,

    242422721299304692999899266957661426754348507,

    7821214281246544780155473195303752183200128192,

    231322293917158261503853547127490651125883514017,

    6311876529883296839313320787589100946109434181056,

    159782928660217826066010854126789181341306883954968,

    3771332416538163680554175720375441261228737745014396,

    83364808207505714122202099093913568600487623058295953

                                                                         n
      Theorem Number , 561,  Let , c[13, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[13, 16](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1137409, 6050460728885, 831561553469123972, 18513641882744202057962,

    129973060893182637594775850, 402220039022358553067237198796,

    668516090280801101905870006884965, 678495839970845358158233833091254449,

    459701836109284848350503706213771569707,

    221870500420639915068201398405358020859599,

    80118444836761040425738533485811075355721150,

    22488491007985633483118285011588504185124394785,

    5058181942184608619559729529421976017371845500059,

    934469881207379761449523794535155383132551799782696,

    144721053808750287423830354041051012473613447021967940,

    19112169259735329346326578681496606191966192770897840800,

    2183654660428852896298786647218456494531593332819338008010,

    218541601823093025881169739032671049958179261579376597051077,

    19364675011506128236218165785303061361822850659581631010670377,

    1533428440345624537146062753622651978389419622358661420355912010,

    109408182342041977844091228836613502550393183585893007417766356026,

    7084521798913066469380956527349821852201007600024753203878480352890,

    419022826853954565221027571344129286424718938983621511181949683038674,

    22768057509707859062185410434623194673561886505491309220338677788601383

                                                                         n
      Theorem Number , 562,  Let , c[13, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 17](5 n + 1), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

6, 512918, 1872386468, 1760598503449, 726847638617036, 169354916338540362,

    25697707490106701268, 2781753815216829585486, 228658032924139402981894,

    14930254895425995405062706, 801032962188285633908585236,

    36249829905082466115817301175, 1412714933662783805918165052500,

    48216899699033148642706180542786, 1461349986381266921516293888029444,

    39787235500118154491624148273101726, 982692870143670903589335233390873236,

    22202928705473751397762136343573130082,

    462229722934737804627719558170397560764,

    8922590903267772214528287949150289900775,

    160584665589612258117066903111012422769570,

    2707766180394248319823833683659212519808982,

    42962543280306339566044395370275707256844400,

    643898739478040894974624731888808717255494175

                                                                         n
      Theorem Number , 563,  Let , c[13, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 17](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1100, 17035846, 32735186596, 21244462101084, 6823854941366900,

    1321322996821051361, 173593973486852311226, 16725660333007373519002,

    1248031861408781862573160, 75065013266877586011692633,

    3751752142930148606295392250, 159570326993208354224507962642,

    5886578313129568266108519239780, 191298312782364603911902074290195,

    5547336088076984611319688030391748, 145102659469232171993709949938009754,

    3455181548810598164480239411353989464,

    75490712491265532347213838822359997930,

    1523729566625751096852315975613798523542,

    28582648817087791925334417433767684373404,

    500902960637574480655448023982316968073592,

    8239028581527045382414796457904409178357046,

    127721291417262827835909312127190574752044762,

    1872926141066857948060690267321680903340042274

                                                                         n
      Theorem Number , 564,  Let , c[13, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[13, 17](5 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

9979, 86964094, 128238004894, 70726392145888, 20277847681047755,

    3605120387359311898, 442799037308560928254, 40383367012473271729080,

    2877965914728742963492352, 166442441140959435463482546,

    8040695979768758637105205188, 331927928921421707248938834000,

    11924605653365984368435474680271, 378427009454711099114762204692024,

    10741094156184448860610835872020202, 275537817608921581466497974489545808,

    6445318011495014195948217790837077276,

    138535560039288780555150820543529796220,

    2754326437254047589952576572756207325358,

    50948126504521889502224365659581755861432,

    881292402131425899209748007291960174464120,

    14320506657487597683791398645629911261737642,

    219480933398839734776448954640728712304261658,

    3184242535212555265334734854090283209061942328

                                                                         n
      Theorem Number , 565,  Let , c[13, 17](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 13
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 17
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[13, 17](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

366370, 23382276140, 163911882043867, 340631665422897840, 316285026648972091610,

    163327166374385287844210, 53617866619198275722637595,

    12232072311890904828963504056, 2065165901052633810611213346982,

    270305006753365070796258717637160, 28419453928655824636874391623644090,

    2467986820578998688914456722395706760,

    181028773968302726913168683282811077402,

    11422674130884992801035300159161492846928,

    629494572951018499435534290922828696045800,

    30687530914504528261460282407339790897746000,

    1337804386476327105757635905229772073814315910,

    52642436598113918360771238605718946460188927892,

    1884973471620306121775323738463267741334453913031,

    61854385675373498743166449434895901429462441470180,

    1871688607792219441615289784812015827341063651946060,

    52515489486386954259840329694317952651331313907782960,

    1372977395409030310623750218292782684625018416727139466,

    33594638411194955150041417980286161434158733532810396600

                                                                        n
      Theorem Number , 566,  Let , c[14, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/5 c[14, 1](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

142, 115096, 17842781, 1230891952, 51102643414, 1483769726210, 32896083805676,

    589402320266099, 8875312481287692, 115558235706257738, 1329096064089824058,

    13729566584947844548, 129070312353724733263, 1116097341036575573844,

    8955779117693784196678, 67177064616365400829944, 473972754057216696432524,

    3162327345215814386003116, 20043395002548394984809896,

    121165221122238121385053508, 701042373975733664410774028,

    3894116883003357019510375768, 20823827342636149199034738985,

    107463184092331478382145664492

                                                                        n
      Theorem Number , 567,  Let , c[14, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



              Then , 1/7 c[14, 1](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2024, 5031046, 1917426718, 285672244335, 23497202718340, 1269631956938732,

    49925328385885914, 1526789932344949008, 38021101110858808622,

    797212386454696838460, 14433628247179449035375, 230107096597883705760004,

    3280922649506448205673776, 42369442139713346665498210,

    500744552839809760293410020, 5463376212146736479895831026,

    55434600179970590492895076840, 526388439039479799299702298743,

    4703207319175911613218746764840, 39727860326529176201774954802280,

    318571924017028680767609779251526, 2433984808413064772785221757084440,

    17776041356849848478634563634877606, 124456233733580999888384411581479530

                                                                        n
      Theorem Number , 568,  Let , c[14, 1](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |           i
                                |  |      1 - q
                               i = 1



             Then , 1/11 c[14, 1](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

447578, 5497765770, 8196341559414, 4034232946039860, 971593593310422096,

    140371632655989268129, 13743042888988457551040, 987588904398417929462314,

    55075100510108236993206140, 2482572918609091332429205488,

    93283845568418857637147137180, 2992950112202852844775566123080,

    83578759811136962627714662849754, 2063248513957828815112744186172080,

    45608366224084919831042707360686788, 912524607880613644295387447532402922,

    16676612993153563316946111502182895321,

    280556870212041287571015933102670624820,

    4374309977191315734013362803934369302720,

    63580240994467821124433230249586239671132,

    865953605566549507741207772466034680464056,

    11102007266455168079098357047823443637926496,

    134523145798905623655979300446540476082270000,

    1546136979301067758055295956153332982374185850

                                                                        n
      Theorem Number , 569,  Let , c[14, 2](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 2
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[14, 2](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

46, 6480, 309302, 8343648, 155508254, 2222717760, 25938237606, 257507522496,

    2238377475342, 17399176546320, 122892929617900, 798644991036960,

    4823280223603200, 27291019901931504, 145647191650600774, 737283672738833760,

    3557062136140144494, 16422921399509398224, 72819749212610548406,

    311048787852317390400, 1283411605562668039996, 5127462039055023653040,

    19877582614288627182694, 74916368457426338787360

                                                                        n
      Theorem Number , 570,  Let , c[14, 3](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 3
                                |  |    (1 - q )
                               i = 1



             Then , 1/17 c[14, 3](17 n + 13), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

20760178, 20828488316216, 728158860798530134, 4794099845454025084576,

    11018171062414542806266250, 12047307699848198142483734854,

    7516587911973570432940827355602, 3010335926354544821788504022093133,

    839511971434939907560948861447716506,

    172952196339503229797826046461579768980,

    27520362857359626420934582013564347747998,

    3501207349271423680092475340600038767089450,

    366049791377055170720315347514947108023389776,

    32157487521743215242216048203085778899456720528,

    2417698664870513145246192609493311922786742166634,

    157960874501300904273147203642437758990377818553938,

    9085472953364379143765065022914053242147318253172796,

    465160467552419155343088961588980283239026621019083734,

    21402395273279964479443147516900833207929595063689674800,

    892347773780264813537706811343338000737352609062065446250,

    33960977761678118383393019021114397039837371246318997264442,

    1187381906782677321471626357580237145688266356783821511004842,

    38356703792517817064378495468569386876172774599377900681697788,

    1150649117372414854538905745686468308090991604070803227285896960

                                                                        n
      Theorem Number , 571,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 4](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

35, 118932, 45724385, 6504791808, 503245643791, 25478620606208, 938851078104065,

    26948075428195922, 631222258870192319, 12478005583509009166,

    213479396963621222348, 3223098471480418519230, 43611513521924285907200,

    535499472864956893459642, 6028443262324820240930432,

    62756577723096844536805678, 608504521114627399407247872,

    5529713926358085666227401040, 47346705788051201752954265873,

    383739981963868134159557232384, 2956017719544628762278992249597,

    21719718221438175495337772375020, 152706619158657564010993701148672,

    1030264635397388471855254152837542

                                                                        n
      Theorem Number , 572,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 4](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

246, 437823, 131183360, 16173362212, 1136527697368, 53626317082240,

    1871164000464878, 51403052618092785, 1161119085205864526,

    22259051490985251712, 370891370365950734762, 5472134396718598850311,

    72552344216970634375666, 874856306034798137755131,

    9689633263171306042322600, 99393108691618419763127089,

    950880511981640728659833088, 8535321142670544448080550465,

    72258128452665813557950206464, 579541743178438917001629572525,

    4421115020661063132684562217148, 32191782589721229091977048658761,

    224425418161520662954560771282450, 1502161399267326348316662844574571

                                                                        n
      Theorem Number , 573,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 4](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

25, 1079228, 1824691713, 811805498120, 163797746470058, 19248625305854230,

    1515665441697919124, 87331508159505387784, 3908667426227570607365,

    141815701636275431705226, 4306030901660585886378880,

    112128363279286248817469170, 2552844234783794719013221620,

    51612948894761295398535861760, 938548135864242326285149639090,

    15514084443884411068098408839300, 235207563218488942235607787415015,

    3295655361798310382989327303163136, 42958191209031650651642602872017920,

    523885320052703714561636132375073572, 6007241296512942883867943115362380595,

    65052977520109787035846849241556602112,

    667883138853032275804244293866773626911,

    6523479639426473075527723567329310235070

                                                                        n
      Theorem Number , 574,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 4](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1001, 10958654, 11552401580, 3953130740010, 670607912931475, 69492566622342400,

    4967676910455508175, 264922407404250524830, 11124526467053367013827,

    382499623474969209614030, 11090480049061907455819264,

    277426653406695285811615010, 6096657959050388891486107475,

    119440886696309429334769404190, 2111441228246163401627851606855,

    34021451655870707718381304918978, 503942443339384950160328816389760,

    6912436232739611310349810053677560, 88356107105921282872845336395587958,

    1058217741586936923139902606558174820,

    11932477809498191419344267334318434923,

    127216161761051936101780839854849169664,

    1287189256445618834058918818906613568475,

    12401894459592623718326928235677602532794

                                                                        n
      Theorem Number , 575,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 4](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4902, 32660275, 28097387652, 8538404815249, 1336545714617770,

    130543202547973666, 8912861131077863690, 458037910845472213619,

    18652587478887386846720, 624897361453427241253665,

    17718058201548899775016494, 434646086510448142433748480,

    9388427844990622689545712734, 181125214747985756510165670360,

    3157939300472187951917544440820, 50250888949256144061514668855987,

    735903310998134622753752966312530, 9989295306543188221449643949133707,

    126463156412292155909912097266358016, 1501207297194719906450126165199879040,

    16788471829681188839527096070366739370,

    177616279257425478177508319658026778880,

    1784272053223137161573816780608510038780,

    17075768495466507557200180086829264811319

                                                                        n
      Theorem Number , 576,  Let , c[14, 4](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 4
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 4](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

21376, 93702400, 66952797919, 18199014718720, 2638579615645952,

    243457685501938688, 15899322493560894080, 788191473684396601600,

    31151081087088775648000, 1017447490884960798608302,

    28222902142949263610223616, 679200365701171949042737920,

    14424283339323314282905368960, 274099987117048667256826594560,

    4714327432119944066705770113280, 74096815060175995434733290812160,

    1072972428048090248797616317553265, 14415213479873052922128945935929600,

    180767967840897892271575233530295680, 2127051448008673856898021230604037376,

    23593926236646489775508700052877147520,

    247721947385270374593411886867795752960,

    2470871760455470896457084143433161047680,

    23489252181341099167866464082349510587250

                                                                        n
      Theorem Number , 577,  Let , c[14, 5](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 5
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[14, 5](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

65, 14801, 1077281, 42528802, 1125238402, 22296977907, 353875545281,

    4702503109762, 53978837839395, 547634420749282, 4997018448528644,

    41573514503737203, 318828184951183683, 2274067713444553491,

    15197250421492719044, 95749276639752988179, 571746360094761645442,

    3250370492432433557602, 17661385131892654492038, 92037287816416059052885,

    461376349730329846009443, 2230767331047364709139445,

    10427700336046562920801443, 47225630227438098452612245

                                                                        n
      Theorem Number , 578,  Let , c[14, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 6](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

336, 933539, 407510208, 70106075648, 6669695317888, 416467589824640,

    18887204428719292, 664500185414231040, 18987204293217848416,

    455587724636570222144, 9414515560780322125056, 170874513990300955564424,

    2767024975898464996401920, 40488511178426537717813632,

    540999163890677985005578240, 6659289156514115293294642176,

    76078669161033517908032089952, 811846062647931794491965631808,

    8136840442554348595630980731392, 76965576597332376509142788897792,

    689966294986647872619708280415616, 5883942008980663998766164248845386,

    47891286782857102465392782124354048, 373144942596357386785203700327165952

                                                                        n
      Theorem Number , 579,  Let , c[14, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 6](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

240, 8788744, 18893182376, 11219986180914, 3033850367839488, 474642989581593600,

    49294764840559688064, 3710027911257496520798, 214875544838059036853312,

    10001096552080846898890968, 386427974207627132146841600,

    12710329675577110044583682578, 363039289745704422259899567104,

    9150508186095809164190143947584, 206241700495611048408312712954624,

    4202815720700474284832974463460990, 78160185295863441529622396616060848,

    1337117551374458561426116574201057608,

    21187174736591888369593630871370262376,

    312814866209926041993736323076481989426,

    4325961168899446584283834711748804967680,

    56293815580421521684404467734463051820032,

    692146295857673087872006443236300244308992,

    8070103282242261158640493260648670825905280

                                                                        n
      Theorem Number , 580,  Let , c[14, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 6](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1506, 29467296, 50075768320, 26013016102208, 6431390463224722,

    942458208973611776, 93054244471145151840, 6724653971985944375040,

    376612398691262914548418, 17038465615079057698418000,

    642472220527574964013328696, 20687200547188397951009944408,

    579890044748522710351404022720, 14373853203061779815997782258432,

    319137506575872289502862475111904, 6415566046630220826262487301397632,

    117842127337326712847537871751035870, 1993231077246979995629544999871709920,

    31255499475745683981678984779016423424,

    457034199802771708682319812084822211168,

    6264020388107427324734959256005350820270,

    80836436160211915431487044236072172282496,

    986183464863358571415829860413713746736240,

    11414705679653445773649646476009133935826304

                                                                        n
      Theorem Number , 581,  Let , c[14, 6](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 6
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 6](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

38544, 291078720, 328633018206, 133843206974432, 28015009728486136,

    3628833161846291872, 325419803311835872960, 21755506378856534933888,

    1142090629401419710440448, 48910404569312667786720352,

    1758995103427273195699064834, 54341906543595369934308635680,

    1468529835277206087491836376440, 35230630035848920748684730965992,

    759547209924168828216940847471758, 14867505003848161160703757436741888,

    266532101854540990560859785947975680, 4408896592547489476302017891947473920,

    67730407102313443082572123863250662174,

    971755846196557811237408720025324328960,

    13085780878907767401170895438392125548312,

    166116092711265863851084640601493810442240,

    1995647477424562620477383427254035029154724,

    22768142671406050110269442031209603435713664

                                                                        n
      Theorem Number , 582,  Let , c[14, 7](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 7
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 7](13 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

7207354, 5807206740160, 197447846550856426, 1341836297274308848375,

    3267714127534587917060084, 3836835608827320937086075652,

    2589637913477438656573670519646, 1126587504700069746811837195381532,

    342063580735364918161978097615448384,

    76815672740622123892232567330907220992,

    13329987726170416042157386513099713160340,

    1849502180801849221066465411952435762064284,

    210822547648908052794987034721591804543505562,

    20183190978414695077244373589554666488865799690,

    1652620766129619309689460561892541241008904982296,

    117510768722863374380656760674441131106056039129842,

    7350240404522040477864918925586379794072490511943235,

    408917786422614077435764060350302196933311109914718852,

    20427700623581145869004690919771323997801844475475480168,

    923966175716075831208904567298299968925317704575128962258,

    38116021087469576488850132677303426085668790756029369144830,

    1443331366375055206085877330155660988853507740295092657670532,

    50455895232667077741283315403743041994031921490995595819522924,

    1636667801824569945962958710981257770199303646884230019888636572

                                                                        n
      Theorem Number , 583,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[14, 8](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

87, 29568, 3068736, 167167866, 5954798784, 155822520960, 3214920727923,

    54812712519168, 798259935669504, 10175679881130786, 115672724713372992,

    1189853143477019904, 11205716968234200654, 97549839679680702336,

    791229788641542950976, 6019594476979775349504, 43200257146932402308544,

    293885732984144377137024, 1903181260398526869311649,

    11775943599397404745452288, 69845255298262645894984896,

    398248175432069904977290002, 2188566204986259219985989888,

    11618533520848559559178725504

                                                                        n
      Theorem Number , 584,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 8](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12672, 245382543, 514127599744, 342111401001216, 108949251009206144,

    20490732204406601984, 2579826204419903721216, 236009094382795804376064,

    16611459638831447825730122, 937956944994111094279709952,

    43854233353204326596178934912, 1740280665294766332964965077504,

    59781040544574335149104162435840, 1806355109715492097214601354255104,

    48650757074510945738238830907153024, 1180978321530719136230131580367061707,

    26082325185928548438885915081877569920,

    528328119604677509431098062576345504512,

    9884195607889190012859636587062853433984,

    171829566520644788363578094445382222974464,

    2790553769264134085461858615442742582556544,

    42536972595888094101666196676238089071656192,

    611148767366792318316776568529661846154465074,

    8307276164916546868009963031695767142891783424

                                                                        n
      Theorem Number , 585,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

202, 1624036032, 90574980381762, 691681369869739132, 1641707584630847822592,

    1757897229115122243768382, 1045139327437209221207513778,

    392928971768630968310734761792, 102001637513352043575268106868884,

    19468626072462294506324419628698686, 2862873326774555175572356667296192128,

    336208803384794778728880585275856230144,

    32438149774677284029649628564668510281498,

    2630549977985697449516830336088649174864588,

    182682108020042064680998716052731029999946820,

    11034555542181527713327270786963731530345815990,

    587368323136222039487804587266641855720103790848,

    27861769822875568588359507953346804675595321821982,

    1189100062119317358565867595965998024275932925697536,

    46042368105140448102269022757918979588140371837059136,

    1629260721924917207487387373200073692564404552056561920,

    53027794917317303087642996808494004414590829286881255934,

    1596487814891630409041568127499453039322093999924245300274,

    44686557413380907034752504513040465733223069375633694847650

                                                                        n
      Theorem Number , 586,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1382, 4968970042, 217707255182592, 1460507606812188946, 3186451186200673516502,

    3211620981653837657850624, 1823169365327841753848813742,

    660645941322161102116836223750, 166386733009892209189249664598918,

    30959572683779692742515619668188288, 4454476803066131868231746176735283054,

    513306531822611593726103221201239278976,

    48706331400790178444401985496742163049692,

    3891721460363716182255188791641405955509760,

    266699299647048369826964956743682393312694976,

    15917267988975452013367599115672422739951041986,

    838078120815954248866598412804927343584140194560,

    39359340241986740033445125462084857923922780837332,

    1664457394139537044419517234972588112438172134138446,

    63904616584340917711278930517096069079516553868735218,

    2243641048630005186335989157411357721927612472871713536,

    72492208397552161425942223195166384779130706589782319818,

    2167648296871419153042115682690048534134716033121099569964,

    60286999098782467746334718070912822727010309614304942479104

                                                                        n
      Theorem Number , 587,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

8064, 14739616092, 515621036913162, 3056104627700236542, 6145929418304084199554,

    5840026509432794704251718, 3168690960231425334321470592,

    1107451332460305848250432322048, 270737331872943252191177916498514,

    49128637096487615186160297676226880, 6918254665929549602464156218843879338,

    782435925960530891038766957274809814444,

    73029609408265034093654093101008363998754,

    5750250951715080024029183397944882028268054,

    388912851960686020747039634518875720280114138,

    22936805388839310620133661368155091178471267776,

    1194671716376234496526725535064620660687991766486,

    55553437407662215523408867525537779766896780120086,

    2327986437761836311584239117824372307413926809120512,

    88631089572651817716055445327620353231741395483176830,

    3087579655437627423819714155279435922685380000691182180,

    99037861075474804262096452302289344328265955107023606272,

    2941386952650646245901635093393557205886493994497363776384,

    81288044070318305981322996488775133810885756668409872574110

                                                                        n
      Theorem Number , 588,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

194426, 119355547934, 2775185422126578, 13039556857349655808,

    22452451756290970033844, 19048705990435267062268608,

    9469190256418039819621832034, 3084819063294136659482470380502,

    711585356157921696246931589111808, 122943364598266376677100281984726090,

    16597843300136349006563764143012999040,

    1809411483991709422304453931112092191616,

    163495053921377078312642711144899059256554,

    12506849245607508345427010265458891356217646,

    824223169300266564718444966061349114422374272,

    47482019595761485872100361935871780178202327740,

    2420806943743133179938007310856317686057947308730,

    110387678691656022167529753390556132554375520916736,

    4543246845311165194171416575698108649318931323537938,

    170114108657936785430126742538257583624828092230636956,

    5835277021733189724834137898135898026748527868056694602,

    184499176972943938470731908694165795083837830173212935262,

    5406365170910363289013074082115988993665119048068858144770,

    147539027917545135276491238456767416996996966751083442763840

                                                                        n
      Theorem Number , 589,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

836928, 327172108928, 6314884078995879, 26604501730822009728,

    42540758142117317208640, 34175606220387680291223424,

    16283728775093815094638879872, 5126514146374417482204464263680,

    1149498706182585880045655407253248, 193892167872707410203404921820696960,

    25640484319413174704541839944311126976,

    2745170963568839213209259627147798068992,

    244123371920243088268961061598487835361920,

    18410877466449470409139593872178182898357338,

    1197886734917686489815526928978795085675537600,

    68213219380689313895979704177255700005118573696,

    3441233203351536405326561357207143193595962555712,

    155407770709080467130279986277874043225472090271360,

    6339417322372476047952331167491007414501323801760960,

    235420902562518928991657830868780466077427622419452160,

    8013897708758637111542549189371060765383349099702417984,

    251582875524498941095293748876156952315132324774320113024,

    7323155864099772056992656911794790591406556510590260663040,

    198603490837057821783126373190750112219668603186649888874880

                                                                        n
      Theorem Number , 590,  Let , c[14, 8](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 8
                                |  |    (1 - q )
                               i = 1



             Then , 1/11 c[14, 8](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

12706894, 2300730226372, 31547106740010816, 108200508969068497706,

    150187605516324602784768, 108613138754200883175624546,

    47668332511062892070587195132, 14040761213614249575883237848314,

    2978741118841171027883375555178432, 479368940762729565814978454115653760,

    60871796403406979532337352210774378460,

    6289942659565848190001586919039997639808,

    542067675642403221934468160616822200436062,

    39750746898495719062901192902738870615886050,

    2521904102533295100285469199169333682032821142,

    140361543484447443662181831138069531915917855290,

    6934803115842237120312395512414724966671244326796,

    307245339211247401912306718217159060364721991956352,

    12314236258158844658564006763364486972632783097183702,

    449904232061144729505449041467651419733777021599027018,

    15084784312257229594666167195762416416034939452815218880,

    466920349595872625018424676674091663177910193432387740772,

    13412944514656295910445833489486542258261305219129148275014,

    359280119952009825642551196002918116518108977818230648124986

                                                                        n
      Theorem Number , 591,  Let , c[14, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 9](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

57, 566513, 535481615, 168316839764, 26724841724646, 2627829087657029,

    180080592971646907, 9278904617924432458, 378797936157159269249,

    12725149090287391015728, 361954812797196942317678,

    8912352764953424101906051, 193335738261856884194353859,

    3748017633536462973626892369, 65699229584252166014772484781,

    1051600300177458791591305024650, 15498152100819917136211091460875,

    211803408426834790534737601287780, 2700680480821902382256573173208849,

    32301251227327119246181920169820441, 364086758597995786835020203914956296,

    3883516858334668513306647159253090884,

    39343677331851331482107701591145718933,

    379820417031273912742409308814555688446

                                                                        n
      Theorem Number , 592,  Let , c[14, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 9](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

506, 2524227, 1813230553, 485722010118, 69198028327063, 6281542383055655,

    404518410596720057, 19822980253601458808, 776222119493490961282,

    25171023227457328722618, 694489117630994447703937,

    16651048839960571884893138, 352808836424919100558701900,

    6697376092547691168928169269, 115199851319693612929489669340,

    1812591432868763994505969423997, 26299159251425259793061513097167,

    354300102685157341714373634460850, 4458380340656081474621638891929253,

    52676472201333180465085039379843135, 587046435432326673073406284844524171,

    6195793463100548478697700471832929735,

    62151049671905152668853584523335576908,

    594456692961008894284914063663396634199

                                                                        n
      Theorem Number , 593,  Let , c[14, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 9](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4030796, 10278785278710, 760232541980525651, 9681162779791280277930,

    40565955353954847419679136, 77414841148276067926001722454,

    81462849394936457897976000495300, 53506781591859686782713127395467380,

    23904249873809674103405224816667529580,

    7731885517555782636284185895787246590580,

    1898055618007290413488263576663988113773484,

    366816016915086964211822893160257924213585910,

    57458424786947785056376287949934998002054967880,

    7469419667837843985048969342597499634887691165140,

    821693421816930654918814221301421475287916219925330,

    77749795184278803277570819430891635930418823327167455,

    6415720314374237773485744567333383348982304282267953590,

    467167640869730534523994121720368863138816397894110677380,

    30325360572648105186208163355686603674442160170854573179470,

    1770505546801661959080231840844963963688238390988058624450516,

    93696909809288497504501479065315648953570401829607059531364244,

    4525564190185536501078150791986279336041918216576249822254844360,

    200719201263593698598914084961824619062187958000050840098831735220,

    8219390587306184282166531983648935847454086309121351768048266413500

                                                                        n
      Theorem Number , 594,  Let , c[14, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/19 c[14, 9](19 n + 18), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

127821581610, 2441817004716955910, 1249002703460139448756020,

    92223866239295225022825364001, 1931984561041832272583983901232948,

    16355539387343461228645680137719888660,

    69246288045747794623395947088685592084520,

    168719407769959526824768383226397629407797170,

    261023914935009761126329269955382850319111156660,

    275585784424188205805950764895214845246098389078926,

    209736264646114381779154023487341546206137734524299210,

    120087360335707046771738522290855367644605738362391625560,

    53525155801546913223012427353462748863765968877659902756418,

    19095265426602525583552533880746626011459850276732825146498360,

    5579332499473836605808040931215803527659743535224282492178982974,

    1361116014479930571048590323022622611906983727667547767236142355180,

    281813618952379976378058135450727330691950995153431082972274638141300,

    50218689565587595420972585272325155777104439331076181398007161704346252,

    7795889753732245081989530779749304718526857400564970475843510327949874120,
    106548899796758768529345612067588080297657714294307708882021714199776940\

    2834, 1294024039681902437976601589509406617492085605608199112764168229778\

    59270812090, 140800808571258852592097485655575186703838621967308275637289\

    02314283963689859720, 138261664319456684704811525626732991161796035876475\

    6895150215008341728939251861793, 1233276813044139565289932756687871728792\
    26675951761344479670835322888070982862144890

                                                                        n
      Theorem Number , 595,  Let , c[14, 9](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 9
                                |  |    (1 - q )
                               i = 1



             Then , 1/23 c[14, 9](23 n + 18), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

105591741330, 39874735572454868034, 139348689649362170015181782,

    46044219223224954464073391584300, 3407587575767645097427508000534702529,

    87341372516914700889538893703729872498520,

    1002811787228489519037767810566659649531011580,

    6095529516840488746572550286968038927311634398504,

    22025434789264369079215763683998058802638781677678780,

    51472999660992669926129208924718031244948044436608302600,

    82912301061879678458683977481997892986543756285903626003940,

    96721378675409014097171950990041212188949273147211530801414402,

    84991965637670349546734136142433402986652680390327902969068377244,

    58082677543178832628550700441783541490848201332522931936677836180620,

    31692674226394149184871648381730474683093554969867063649095926521285380,

    14114736211592363928401699124385276870712575631139717991126755386485476030,
    522730914041899649521181174361582254877331548746234379834659688319987001\

    3472, 1635667453019316117158400976658548562845061020142253487757706525262\

    814893134244, 43842724088823831526505653708923621774260982406239863969930\

    6424116130499487556580, 1018793889036463119152553146829111428132742105688\

    46328048423733527603189072799163170, 207406308529642797801245438536534911\

    72079835683013493703869970460565465063796751246300, 373362113400135262342\

    6670014993091202035791181177951748952332485512159451041743998746902, 5992\
    199927131206887172552950669150433322434439897747663396560118305637191286\

    48848658943428, 863738450391556957638171647049395934059739990039650165478\
    21821864634508377157747287885239510

                                                                         n
      Theorem Number , 596,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[14, 11](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

112, 53760, 7592000, 547642480, 25305314528, 845077866560, 21956761753472,

    466178419800800, 8373935729241040, 130571202698306544, 1802308098827571776,

    22365087076202043408, 252609415070873847200, 2623396915878454410016,

    25261965758670548899712, 227160920536678578777376,

    1919026653656422579861488, 15309468878195044619211600,

    115857346721433830725410928, 834984859953569184384587312,

    5750801655830833935758510176, 37966876622893355333951235488,

    240932273217459242252630725600, 1473200771392985218446553271200

                                                                         n
      Theorem Number , 597,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 11](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

645, 4555200, 4407131680, 1539928255040, 279707051880480, 31814009661046525,

    2531685308573395520, 151565649042524308320, 7181185963819340353120,

    279444738407727305720160, 9185681326917026771526960,

    260718233861545765964467200, 6502573019221779485441012160,

    144559363930475545351578435360, 2898524726324787701367420397120,

    52938683937734771544576312874205, 888136267522277456068510685739360,

    13785468016534439592788058560118560, 199206169345457153665815837390299360,

    2694522979058847550310911058170555680,

    34279219746481179060559343810087340240,

    411890935600824688917676837554640708640,

    4692082966540822984627808993737846781760,

    50843893287343903999175274461972099002240

                                                                         n
      Theorem Number , 598,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 11](7 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

48, 14384080, 115994684784, 199790751343200, 135951443143646160,

    48726438565439532224, 10826556753715949528448, 1653452557792455712234720,

    186227309901104118546048000, 16271518552668580857407672352,

    1145493737644835132771200600800, 66900719196774172904517870176400,

    3317316803861589925835005710199104, 142290120961040824047011312421642400,

    5361194165712830471822383514753131984,

    179724327746926592888275782446163113952,

    5418825881042228934268848985185965057776,

    148307819065020035875987983450993603912800,

    3713962519235193279367003475441689299110400,

    85689590570305440247277947952461196067701376,

    1832590430987408229993298758849385236288577168,

    36523170910879952701839236576387804186576353104,

    681545199029672627101810076739233933218902444608,

    11958577754185469201552948555106761252015877342400

                                                                         n
      Theorem Number , 599,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 11](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

3536, 234703920, 1099948753600, 1393614346473328, 772417756640387904,

    238306166360577731072, 47155845001415221470720, 6561200947797876265376400,

    684182865554095737848850848, 56009683259073513344749279472,

    3728085856547535833449065965472, 207341083817066997667476668864656,

    9846762868953171137705756114370400, 406414084358473653739696008883941120,

    14792187745601253807573968680321832752,

    480590702774124472324571547548555294896,

    14082503743587995037346246390583078243760,

    375476972625747124027282495234964312622400,

    9179128907042754418194523529677264150835920,

    207120512104748809936870429420363829595940944,

    4338938034516642677115722273707271559669602496,

    84824658957914937552252627616267243792665493344,

    1554638694860494840257474433510114965579815541600,

    26821549674486746949904013948413892581194558260544

                                                                         n
      Theorem Number , 600,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 11](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

23040, 878498864, 3255906677200, 3588829598246160, 1808346648980996800,

    519870563628667665232, 97354680230005105190304, 12954731743206613742438544,

    1301684445461190485900105296, 103256688521768246679698882400,

    6688588439617377486565557239120, 363249338240675124895450939561808,

    16891746514339680233061375186764256, 684207824235541913547251359833403440,

    24485156961772270757542388435776671600,

    783398122341985928346310189014921612768,

    22636440904912139966275404173847204046032,

    595848471858629277380681981464983772492592,

    14395115610161335559101538778966378441262512,

    321277479726745571112857750887337588386652800,

    6662250529671409899647789023730749166801434176,

    129014526964382490659013765451599151949160230016,

    2343641604605281751835366532714726686298580626832,

    40098619595950165840104408664393461503225539084368

                                                                         n
      Theorem Number , 601,  Let , c[14, 11](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 11
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 11](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

132199, 3147951200, 9410040751488, 9098196073542368, 4185971530545217248,

    1124312963947909032864, 199603384576948075514400,

    25432861229503661312929295, 2464629281070357401039361504,

    189575705077106240328743721248, 11957066801220362679809932502016,

    634383048230198182906079061242400, 28895762081068582210680193733248768,

    1148976193535722123513637122691981184,

    40437296289947768050128031205533094352,

    1274346638185315219609158444365664179200,

    36317066633817074285125196044265785001600,

    943906266357717180631022354948209993765536,

    22538607744091862664466064450335809478117120,

    497606925249951106419850630742207971983128320,

    10215327243697702464309656875093088726708873856,

    195969649674583425636876581483249437510494156175,

    3528761565521892334757367904334699640712729792800,

    59879260707017830901528321029584844605550644688608

                                                                         n
      Theorem Number , 602,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



            Then , 1/11 c[14, 12](11 n + 10), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

53159775, 31251960995840, 1145162971919216640, 9335630373403586404352,

    28398692533122757139668992, 42353374563115575729895211008,

    36541085473415087802088077692928, 20348375587136335468782644934451200,

    7898925409914255693424418561874190336,

    2261964942718447571995850504547420856320,

    498889330873996700207462631313211310309376,

    87654320800662745446847968931987039409536934,

    12604271226931637026564128998284184781914501120,

    1516340856644902355608392869513515590102290931712,

    155427612969453187713771662059169572593188724776960,

    13783461404071793635839359586463244835539375292866560,

    1071351802718852495735545513756906777437287722446557184,

    73806212795870474841557681529206063894500931668342579200,

    4550234443389123494678049272481144765821576533159361802240,

    253170910602928527099640459313409237513514491345837655924736,

    12807050428243145484930847135482392421648262830082020146442240,

    592904373162553418546608313135633137875215721743508570283638784,

    25266882964383841282646443038161537038367969257499997816411968437,

    996355385267549711508947165580714710301295523509608902957021536256

                                                                         n
      Theorem Number , 603,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 12](13 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

278, 102408921402, 66099153676240174, 3647940970418788172400,

    47943317422149379809764518, 237835906520310315852890829824,

    574920357642039984053850642367488, 794289951287775524049560142916145102,

    698463393161878711443465792218684295808,

    422137126124151054021699149572717262569472,

    185655302398730283877418244828836785180969074,

    62079430304360334804734968954266907129724294478,

    16337287365385172904668806985561011311483426906112,

    3479213197396338695931963124695326945651332582727680,

    613401219405948661659692540300256524846376049333380834,

    91242715622728378294376927847910491585813819054465365998,

    11635160804842878990892976969792178528264489299883302204704,

    1289350884738575611948581741922796540236412634806050937743670,

    125624195702472187168896903562203085664324438147335554277787360,

    10871416083901219525626252982999479982986035235485119763973317950,

    843069941380234371276801447799066293331865442969669071732864376832,

    59047214282023602796680600665868315737137478388617712038396088169670,

    3760975757229348243282345147508726428477969294440318168194172316790754,

    219202980013062526474000747226721104909492046372818151040234509320986144

                                                                         n
      Theorem Number , 604,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 12](13 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

88286, 3090760136778, 968984053162414080, 36171573395281698760050,

    368233761894964160955899584, 1519682832185573957919080243334,

    3190759365464286358638885148245398, 3939402908150136758408924969401823936,

    3158079655921259151601539895762130938994,

    1765702646722104069910403012943914360095744,

    726416630286138961772589557357646532663771136,

    229199525364643188034748715385633418428759602734,

    57312409044719885287962665003835956265871824471440,

    11662928880368440768787150419315053322379471401656320,

    1974059407446891826799533436500279858537500219387409764,

    283012814038345271245924456955630625503591240459593067008,

    34899395139658365060820897601990168905666081462388216853760,

    3750535473709175203852761879326896926126444020324554125981654,

    355258077667983034965170311263622635840824180111021301079398400,

    29952980641554330094288349404362538571197250917245855654704293326,

    2267367158768142079028181469601328748555988735733253108003438765824,

    155270109247868719219369621002198820197753272640381668816815630115424,

    9684220791620104549917546256075847382389586745489617546559478375474672,

    553430892902807731262434654260199821571350733285910201495843995473001984

                                                                         n
      Theorem Number , 605,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 12](13 n + 7), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

471712, 9143749838486, 2313461147477321886, 76526070271904249677056,

    718792601314570270009907018, 2797418366375166027420954415696,

    5613680796767345531664265514009294, 6683706116081293279051431090816622592,

    5199823918682598960362502314492429336064,

    2834576941478842943959736668078537667993270,

    1141060486046571987901400195578461071842959360,

    353266161969343827393374000105851715943148363746,

    86871030764131302094098634660848788337595390716416,

    17416702156546660201317990344703810571019429309486646,

    2908766429126309606782983850626744088792595119442605550,

    412000955695436883078209947006928375345550590614126426082,

    50248566055105150601403298321176760222170620674147969735360,

    5345844864830078548477716857106729984853048562489231554625536,

    501688315752929815693283957268612655125182533782968790240002048,

    41937476864643708081061472633913872291941450330407113632060207102,

    3149377770634457255067723932108264083863691291676147140234931804832,

    214076478668514180656255556705722410965875646145954913750802296782870,

    13259751270243529346066204662060467738034922106142313256538109582809262,

    752857617518093466887099552469545422042739322270467295919013136085397810

                                                                         n
      Theorem Number , 606,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 12](13 n + 8), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

2311718, 26443966996480, 5460097153737666910, 160726475964292883539264,

    1395886440239153702722541312, 5129503669511345552782541633982,

    9846201651223252449806125618650078, 11311398284754623006938751787385170592,

    8543656000573288164163734740370193333360,

    4542367099567889557031480794239666025446912,

    1789610805121758684903162496171028097523817218,

    543750685391935506282378194657785138416282901990,

    131515672512614235757806790973143484501928920965120,

    25980975545385448419659706577847255549812034200037312,

    4281881769849833087331193459470536051215737715562244094,

    599247748884014193546550807777033338138620895514582682678,

    72290085832922689160250063651495085056108846606546956576042,

    7614079300673823880174024389916156108217228233437956355564046,

    707989688937608704137259851733227272719857598113623674394685440,

    58679897063861742255577272579881091096462088907825908450093649606,

    4371915489483898154026101823187277529074026357362375915390629339302,

    294992705333555396974984188186283600373036184115105609587034253058818,

    18146070738750401787112380137175447577619724866565975473440277234792930,

    1023652495966537359323068432716370605380380148813782581143165618917949440

                                                                         n
      Theorem Number , 607,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



             Then , 1/13 c[14, 12](13 n + 9), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

10526466, 74867459262384, 12744773575932354122, 335192902467872375494794,

    2697212998369066458440064402, 9369994516932191698225216829184,

    17217856266038437704354545713766400, 19096022142479599610524319054879190010,

    14008771548037340140997933590986351788032,

    7266217898972039776188643233995293272334976,

    2802480300326326409742608339948702188120283978,

    835819846830066307259969010103580684613216469706,

    198866155815441846918034664004371839903169341626368,

    38715069421584897223239136881723502707887890579949794,

    6297104575854389149024866893084524238734585141416483718,

    870830043753515800324392821725094620484019610942038652160,

    103916726727926945711183360168951647708317124995703519899360,

    10836734977744200025710716806946639741394683933146324739297280,

    998446007615368890142519102175796974872280746332391070949671190,

    82054565910977987693610272954116622664952904631575109233513001728,

    6065466996063207313765190858572165030669559314113012091810695908754,

    406271230666113616533335194669243240958160117166531528500540206190608,

    24820324598983142303396438740061549192688609759577271475822558784001190,

    1391180549861689044883895826029750036682022362951907982141758366179015658

                                                                         n
      Theorem Number , 608,  Let , c[14, 12](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 12
                                |  |    (1 - q )
                               i = 1



            Then , 1/13 c[14, 12](13 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

699328944, 1513855568305664, 152302464590411686894, 2919849851532379593843830,

    18902302195972591879937610874, 55862245784798953811702649932338,

    90460120629111566742660763290320896, 90560708217167457417287994712827005114,

    61008257762640753662845556862434418365184,

    29435187296753523829256120984016499256123710,

    10665152576634462099400416844702002507773808640,

    3011596430029643151073581807777409279604484711254,

    682717542968228890232276096504455102584155575778086,

    127292161852129937050333303099474599071851457453010592,

    19914695820997472981803443784549902014282458568602026304,

    2658606925476933196055435830411779991351616432714109583360,

    307212354882065994203621653496275085925610788102541058840576,

    31105830324389593689654522923837268294022755314049639040612642,

    2789095301997200639457012747730844081686088153484634595208741458,

    223520129312287589723003279340795831806507797754091582797218740982,

    16140921580769076976353698269933093858972393929512589135834466659984,

    1057833904641911124266488284677402660943676249319517077423863650238464,

    63322607218631766471580781911079773848452495318351107015465411873433966,

    3482039323387458707845751859663689979232327513959842642464890593387694746

                                                                         n
      Theorem Number , 609,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 13](7 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

576, 115368000, 1003386388608, 2024198568784320, 1651742390157796928,

    714501028750132159872, 191625548228708499951360, 35236226658910047732615744,

    4761111626159626451730804224, 497063393045992739408660619712,

    41639522620269294173708326892800, 2882116843614552124750900304587648,

    168707643964616678332569629111796480, 8510680862083659559749406534984789504,

    375792526158132870924613561983769718400,

    14713851903166201299525378918199540886016,

    516501278462173060800203612584317134497152,

    16408341861033705444821726827378973411526208,

    475586844870700470430607575015807349155612416,

    12665843848445527723816346866638727785580308800,

    311865704871708535926171852961858382614956103616,

    7138471526187281089679021676818619794268904401152,

    152635781216564563414585573505060530261393739201920,

    3061997097506587271216111580331467777195164183614272

                                                                         n
      Theorem Number , 610,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 13](7 n + 4), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

4736, 479043264, 3195591175296, 5530116943680192, 4068588635011045952,

    1630874424581558237824, 412265619880054441434880,

    72275226154831824013747072, 9386883307768133861973861248,

    947681806044322313586577785088, 77127086835041465841761546310400,

    5205321258956608187649392473303232, 297974874401982034336802380732800064,

    14735378963733163523457915181831902528,

    639093469546276703813030312025023922240,

    24620226957521865999115495610702908426176,

    851546494955785405726783875664355834288832,

    26687495340664270026080754525885650168428608,

    763911725328370242973750581301742672265378432,

    20110528070833984214250200189070734678684644096,

    489880618726608126175998380228095025524521911680,

    11101399669017056850929363660471345114900183189888,

    235158576807424836225249312561899363271248519314240,

    4676208620865313549619700761217973484594438956217088

                                                                         n
      Theorem Number , 611,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 13](7 n + 6), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

201664, 7145127744, 29990273629568, 39245609755195648, 23807832476187212672,

    8264963374264588528576, 1866587909385357046090752,

    298620105197116868637123200, 35935688019787644405414975872,

    3400087906165331293966930615040, 261622390053943017368333360080960,

    16810303241011023164441487674945216, 921310553169022089325408406961963904,

    43821708898405491505515025983285721088,

    1835117903409350582692222460569397575040,

    68481472945319451110935289976521000702080,

    2300775309549832144625318190409582129443648,

    70209435626899462606524237984520615268614464,

    1960893383470529870913245609570991223258821184,

    50459942564168081813508879712653372206542518528,

    1203433193360716726943911072725526817236155743296,

    26738396547152801989518069319860650555264129031424,

    556026076706925101151186712841519119290704437489536,

    10866728457632357643382638052526668104687435648758784

                                                                         n
      Theorem Number , 612,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/17 c[14, 13](17 n + 12), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

779067647, 107815977486726656, 169756431715316534708480,

    28738671902309412144738010240, 1186753994429521463132723654830208,

    18044233075814025914035598934294120448,

    128735922148648519481890589990735211012992,

    504387489819697893520587159297771750898774912,

    1210300463264667520662049118832575287073607820800,

    1925497667415129108666935607560342023068298393736448,

    2156370176052719816397892171394414017029092751071959552,

    1780932598318285005291304746235624938297207170789674460032,

    1125639092770578768006734472565935194539873441050774771713920,

    561084467425127670522992905304961444989943559117189541777366784,

    226098823817647727377872119433250855305192122069746644677980531328,

    75199058116180457883895121019271649257442675504231473416386283104000,

    21008384916891320181694312558004829190932564321175338747286049698893184,

    5004498283692103009297698770006877718568213390203663105873171369605593829,
    102978793903363806107739032206751155591310703817944263644321754181073915\

    1616, 1851220340498179761676197293890487794332190316756586718078806685355\

    61046886400, 293624795493480161500271211588070972719527400431551231206655\

    82638324898327928960, 414523815853794974927140648301599356353522140821504\

    0869997411643411734110504039040, 5249305427646469745354568392135324043496\

    30521387753154755516914040483854662494510208, 600430253264176460446391743\
    15252106218996553861311810684365436010295133338675447987712

                                                                         n
      Theorem Number , 613,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/19 c[14, 13](19 n + 15), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

33373204833, 8771306701753183616, 26627714899148566741906816,

    8250146888386940069754139969920, 594254823606108089439613232703035520,

    15144255753219875045699443827070680664320,

    175216206004994910158644896058455339162594048,

    1082900414499965676381833422113356835802701734400,

    4003531537022447552825182440404561933305897344279552,

    9616087253632328462664002717248005667584076874708910464,

    15972801069255787493306211820853398288868065018400852848000,

    19262338495402914056527468432749103723897580775183867349073664,

    17531078631784782817145224604921122608753251236287905862822800256,

    12426478715191284795251838700559944933319497395964308726979275237760,

    7040631657678618986994646385491553997271954276610343338720986991235584,

    3258708670400408595984928511402888622396470146948445578231570398696577280,
    125502608134298511333858269069301265257889210592784792916493411927498320\

    1408, 4085871311949914096516117991063115067935712341949343404775891856827\

    01006727168, 113988539824740259493969908370027565435564580723482148351313\

    437362526430144474112, 27576503961215055982969788278402115940172580321866\

    230506534682683017958156773406011, 58458006750641637000808250824394448684\

    60130740236870714523787630731566655265602529408, 109590449846547295373677\

    7720279505524847571244070583790471623111710725122178843086442496, 1831795\
    446697254037593940021531437665439530956260084771424797777449644847729752\

    82348589824, 274998086936424707888382970611075251952236686960658490130745\
    15226824278369581525486082412160

                                                                         n
      Theorem Number , 614,  Let , c[14, 13](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 13
                                |  |    (1 - q )
                               i = 1



            Then , 1/23 c[14, 13](23 n + 22), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

78100392191423, 58320819026128673898240, 547493644627735195009171103872,

    490906158631132769537071800928594560,

    95152638109870644834440870862717329879168,

    6120595499819038673902234840151962728295326464,

    169225327693411987306882912603940601523257872279424,

    2385146693822659025131654794904906341085872981022023552,

    19314596862920657191986511490262170289979995498451894796160,

    98104492520043830727023666103734742060564807842930176155282304,

    334104928276102360767381575028168109566669722067764613458638935680,

    803692496541761963624070217083920720406666459588164207120204230776320,

    1423596704927294777182456392158067166892046801367749945045403200148864000,
    192084752384814332365633699285538458346290040780218827054516068206839452\

    4032, 2030410053896932830253993446610842778617896054843259115242878608673\

    563131918592, 17213524247496560214435883933243944242685879725965711672931\

    60341975419708660269696, 119402643124861105227845733993629517275528432405\

    3547889018733520325347755951584692480, 6893537763077910375214174623885503\

    30893161531041615446683608752343292823986695305314688, 336184727313929834\

    487964044802519356853857539944540399254570384998231991363742809561627648,
    140288698741134851747966924044752184349936979504904680240063334465110550\

    069528362836609036160, 50663575651328753211989433412577409135287288374040\

    055261549459512150486328887962257381657613440, 15993602357736855563084744\

    188687343291403356539172482626093678176775304335121957759036008584862080,
    445288014615765043222009955044629472605022864678781267899350084288879521\

    5233076166430520228333182080, 1102139989657203448292560441535106007737489\
    577427315199029030290605002173629399891554811331823628057381

                                                                         n
      Theorem Number , 615,  Let , c[14, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/3 c[14, 14](3 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

140, 91112, 16907660, 1566036864, 91289440512, 3792253422440, 121153374534808,

    3132222498397440, 67940440509702040, 1269917235209387240,

    20877997377746423180, 306808179560064806400, 4082602988710118839680,

    49717281936307054602344, 558996243782074874423424,

    5846140621039424009310720, 57231947045618204067847716,

    527330898129915792829136360, 4594706595269858485618995456,

    38015406706200775178035879296, 299759083474935816309274260480,

    2259955228912330002113400285904, 16337809514623745405372971699480,

    113547128265308371916447873277696

                                                                         n
      Theorem Number , 616,  Let , c[14, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 14](5 n + 2), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

84, 1954048, 3806902400, 2275352053464, 649097215908736, 109864469528749824,

    12526798426647853908, 1044445301472380446296, 67397033097154721439744,

    3507684372623654405586432, 151895629929279566121309224,

    5606920281227429389100560384, 179855450084961489785564556288,

    5092542374936398223182726433664, 128932705122606440822963710075604,

    2950536661185809580088837766307840, 61591924158299463299912963654953896,

    1182048350548413218735646172364461824,

    20997962868965063886124019546295741696,

    347306485361368398123959275223061465304,

    5376411927588433492856879033119704158464,

    78253594940356690419032339606433795589248,

    1075264887799705424077544254261202969949952,

    13999272307385805627004696934036133308833880

                                                                         n
      Theorem Number , 617,  Let , c[14, 14](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 14
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 14](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

896, 10144596, 14739214168, 7412039170004, 1879333499038464, 291494848017168128,

    31046477587355209728, 2449561793226071303808, 150973806977065680927872,

    7556850655461756055091328, 316398538877949475697481816,

    11339802660131351007389664296, 354377720457446796720970281216,

    9802685708774247243223783019688, 243021913130253927573253707294168,

    5456352362945492257869474032795904, 111935224199498445061266711418051968,

    2114181038876599880327005391838858112,

    37007406566262735691410643345673191424,

    603812641380972801207203163591159941460,

    9229471105395054331477597979433568947456,

    132755804498093833480069872962864636481792,

    1804095822195189856819649165178403556590296,

    23245576314584448337288187431694941193078144

                                                                         n
      Theorem Number , 618,  Let , c[14, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



              Then , 1/7 c[14, 15](7 n + 5), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

45965, 3859440651, 27940029705742, 55832500119629852, 48392984085587501717,

    22997859277044575568358, 6898047386535321300505350,

    1432580800583262758587365221, 219810520241767914346134500891,

    26133210716195155043768835451585, 2496148796450388153116859801334312,

    197046534044328320501343264213530625,

    13149633762062686421910469066538233047,

    755635177141997503893385603309756910768,

    37965911743296658574860578819581699935362,

    1689361482722219632295381738322822184802478,

    67301261668631595765957606690739651565876068,

    2422976601228122660653285241144357233639787516,

    79471213579977798965807332878502933659940625248,

    2391484303697249008805286723701245529960272520713,

    66437624133679928146471690641769258332921826080147,

    1713289846110327451883535663498358026433047008263203,

    41213238898288442659491829245976146739868303393299137,

    928810992685613301897490263025492255128901653047115293

                                                                         n
      Theorem Number , 619,  Let , c[14, 15](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 15
                                |  |    (1 - q )
                               i = 1



            Then , 1/29 c[14, 15](29 n + 23), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

585731487949775, 77836770236373919324949593,

    25757268082416168161619350116258359,

    407776909622604738829919729939991561848874,

    932574637491081389413522638313276886736502105391,

    539771029123136766694711371042864055874802635881321083,

    110178799564044808197523262740294372244071063238831261379698,

    9842705651904479137407223430446868232066417154703657235019066907,

    447077655074323618265985772987865468159504965509796258934267458121832,

    11517129500535799030717766554616735064116363109340400908948151144177279551,
    182743470060484266664159465463445718543129905484384046191331081935949809\

    048073, 19043640797000644633109536286396340239150250316796323935360355234\

    73230108131569636, 137169896149269549247036038344296204379835231531917275\

    37544741816241989583472763787254, 711835271164597461357948755595928180651\

    70022306134079629816360423964211494777021555423390, 275404988759381779795\

    788603543923985314669683145502377132231326008164731274882762821793047569,
    817461912302091732701496795723506187505136264233135404748530214124864617\

    811067857642307514935820, 19071869970031047889671823676867967335390848737\

    74704485311649802556702233713473621229083307393728520, 357072660685823133\
    499357914093070991311453234591233817307569603808879788744968778816589376\

    2496486964425, 5461813389264049780462131130274615383925912859491324958262\

    086276946026449666072465473495087055264920418138, 69327907983944061089334\
    928992566512831098181384744280030530637560259685883082215930751058841829\

    50720366572721, 740307907653617173121788107420045925792568750492424701462\

    1278107106812496465723135883382187051082815005940012627, 6731262264520851\
    724939141884641163248768787819301093886366253761025521576836029560011423\

    131902452355338880142756352, 52676995164024175317088771581026681032395130\
    557850945330987755378476336164175249923918319296926321052979600221527428\

    77, 358218563894433750676069969364339590581775298678500874616712377417023\
    2636318166008696742775613793628831385647233501722819

                                                                         n
      Theorem Number , 620,  Let , c[14, 16](n), be the coefficient of, q

                            in the power series of



                              infinity
                              --------'       i 14
                             '  |  |    (1 + q )
                                |  |    ----------
                                |  |          i 16
                                |  |    (1 - q )
                               i = 1



              Then , 1/5 c[14, 16](5 n + 3), is always an integer



       For the sake of our beloved OEIS, here are the first coefficients

1094, 16512949, 30775837134, 19355642638566, 6026313732754666,

    1131711605501559243, 144298040190632845654, 13502449081198230063638,

    979167660495284272152826, 57273960895794071434445971,

    2785565151217380549843912010, 115357743643418375623417179461,

    4145853604572472945751314720126, 131324415558246893231479974774818,

    3713777430792588184141007067616188, 94777544615604749162511305107393241,

    2202876288475896719847704729117670594,

    46998297443749083762833605529091118523,

    926694618704722934673892013102180680970,

    16987656574684319326816593286127665027529,

    291032152585284404876708153385758651565912,

    4681300383847002894116677904210566698619677,

    70989950576249238579992639297405313600858666,

    1018663959367275355258920568548972973831649100


##This program has been terminated, since enough is enough