Everything you Ever Wanted To Know About the Sequences Enumerating 123-Avoid\
    ing rn-letter Words

                With r 1's, r 2's, ..., r n's, for r=1 to r=, 5



                              By Shalosh B. Ekhad

In this article, dedicated to Neil Sloane (b. Oct. 10, 1939) on the occasion \
    of his turning 75-years-old

We will find the first, 20, terms of the enumerating sequences described in t\
    he title, and, space permitting

discover linear recurrence equations with polynomial coefficients and asympt\
    otic expressions for them.



   -------------------------------------------------------------------------



                               Theorem number, 1

 Let , A[1](n), be the number of words,w, of length, n, with exactly, 1,

    occurrences of the letter i, for i from 1 to n

that avoid the pattern 123, i.e., you can't find 1<=i1<i2<i3<=, n,

    such that w[i1]<w[i2]<w[i3]

           For the sake of Sloane's OEIS, the first, 20,  terms are

[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440,
9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420]


The sequence is annihilated by the following linear recurrence operator with\
     polynomial coefficients, written in Maple input form



-2*(1+2*n)/(2+n)+N


                      More versbosely, we have , for n>=1

                      2 (1 + 2 n) A[1](n)
                    - ------------------- + A[1](n + 1) = 0
                             2 + n



                       subject to the initial conditions



                                  A[1](1) = 1



                            In Maple input format:

-2*(1+2*n)/(2+n)*A[1](n)+A[1](n+1) = 0


        Just for fun, the number of ways of doing it with n=, 1000, is

2046105521468021692642519982997827217179245642339057975844538099572176010191891\
8639649680261564537524490157505694285950973181636343701546373806668828863752033\
5965324339092971743108044350900750477291297314225320935212694683984479674769763\
8537600100637918819326569730982083021538057087711176285777909275869648636874856\
8059565800576731736556668870034939446501641533969109270374063017990525846636110\
1689727289330553211629214327103714071875162583981207268246434315379295628174858\
2435751481498598087586998603921577523657477775758899987954012641033870640665444\
651660246024318184109046864244732001962029120


                 the asymptotics of, A[1](n), to order, 6, is

 n      3/2 /     9     145      1155      36939      295911      4735445  \
4  (1/n)    |1 - --- + ------ - ------- + -------- - --------- + ----------|
            |    8 n        2         3          4           5            6|
            \          128 n    1024 n    32768 n    262144 n    4194304 n /

       /   1/2
      /  Pi
     /



                           and in Maple input format

1/Pi^(1/2)*4^n*(1/n)^(3/2)*(1-9/8/n+145/128/n^2-1155/1024/n^3+36939/32768/n^4-\
295911/262144/n^5+4735445/4194304/n^6)


As a check, the ratio of the actual value at, 1000,

    to the one predicted by this formula is, 0.999999999999999999998872012760

                                    Not bad!



   -------------------------------------------------------------------------



                               Theorem number, 2

 Let , A[2](n), be the number of words,w, of length, 2 n, with exactly, 2,

    occurrences of the letter i, for i from 1 to n

that avoid the pattern 123, i.e., you can't find 1<=i1<i2<i3<=, 2 n,

    such that w[i1]<w[i2]<w[i3]

           For the sake of Sloane's OEIS, the first, 20,  terms are

[1, 6, 43, 352, 3114, 29004, 280221, 2782476, 28221784, 291138856, 3045298326,
32222872906, 344293297768, 3709496350512, 40256666304723, 439645950112788, 
4828214610825948, 53286643424088024, 590705976259292856, 6574347641664629388]


The sequence is annihilated by the following linear recurrence operator with\
     polynomial coefficients, written in Maple input form



-3*(7*n+12)*(1+2*n)*(n+1)/(2*n+5)/(7*n+5)/(2+n)-1/2*(528+1426*n+1215*n^2+329*n^
3)/(2*n+5)/(7*n+5)/(2+n)*N+N^2


                      More versbosely, we have , for n>=1

  3 (7 n + 12) (1 + 2 n) (n + 1) A[2](n)
- --------------------------------------
       (2 n + 5) (7 n + 5) (2 + n)

                                 2        3
           (528 + 1426 n + 1215 n  + 329 n ) A[2](n + 1)
     - 1/2 --------------------------------------------- + A[2](2 + n) = 0
                    (2 n + 5) (7 n + 5) (2 + n)



                       subject to the initial conditions



                            A[2](1) = 1, A[2](2) = 6



                            In Maple input format:

-3*(7*n+12)*(1+2*n)*(n+1)/(2*n+5)/(7*n+5)/(2+n)*A[2](n)-1/2*(528+1426*n+1215*n^
2+329*n^3)/(2*n+5)/(7*n+5)/(2+n)*A[2](n+1)+A[2](2+n) = 0


        Just for fun, the number of ways of doing it with n=, 1000, is

7593292178303937816728707609005946647161387588845274690037158838707786359743044\
7177063903357683809020622645569077809447613630071717860169753006538076594097911\
6449649663845292417890514498169434155891303506029978474229706193197142445784355\
7757891920617817019606240089635383343807973904796865989763712497079030149944140\
0190106054807028676976232278444601268806545873001139422356155041251677905408231\
0161961984203594782322866051469424736074531828620364999575605087637074413359223\
5157925345436967595919379507226242221927191329554350768160441177225519485789363\
7445372630446831796201611624095188175015347404572835611837121937268895931965899\
4583868742530801201412409606953182855856074434940236704800846830693295661785237\
3524181602941386412189160857694969557657005809854087923073329654300668751515464\
6295533359421150422232691749581601158349757050019014608078466177917132484082461\
1552626198262050573307970487425689073324264845414850506226680159765044693128127\
1443016750250010726630469084361959403774424958401426816024967042779427538065380\
81085581790684652924446696583829800892890399436


                 the asymptotics of, A[2](n), to order, 6, is

   1/2  1/2   n      3/2 /     249     13255       2674485       2123737437
3 3    7    12  (1/n)    |1 - ----- + -------- - ----------- + --------------
                         |    392 n          2             3                4
                         \            43904 n    17210368 n    26985857024 n

         51537243879        30228540821765   \   /       1/2
     - ---------------- + -------------------|  /  (49 Pi   )
                      5                     6| /
       1511207993344 n    1184787066781696 n /



                           and in Maple input format

3/49*3^(1/2)*7^(1/2)/Pi^(1/2)*12^n*(1/n)^(3/2)*(1-249/392/n+13255/43904/n^2-\
2674485/17210368/n^3+2123737437/26985857024/n^4-51537243879/1511207993344/n^5+
30228540821765/1184787066781696/n^6)


As a check, the ratio of the actual value at, 1000,

    to the one predicted by this formula is, 0.999999999999999999999996653790

                                    Not bad!



   -------------------------------------------------------------------------



                               Theorem number, 3

 Let , A[3](n), be the number of words,w, of length, 3 n, with exactly, 3,

    occurrences of the letter i, for i from 1 to n

that avoid the pattern 123, i.e., you can't find 1<=i1<i2<i3<=, 3 n,

    such that w[i1]<w[i2]<w[i3]

           For the sake of Sloane's OEIS, the first, 20,  terms are

[1, 20, 374, 8124, 190893, 4727788, 121543500, 3212914524, 86782926068, 
2384725558736, 66456350375566, 1873703883228900, 53351152389518550, 
1531960347453263112, 44311785923563130392, 1289909841595078198172, 
37760636720455988917420, 1110927659386926734186992, 32829851946481173252840184,
974077753606734750485658224]


The sequence is annihilated by the following linear recurrence operator with\
     polynomial coefficients, written in Maple input form



-64/3*(4*n+1)*(2*n+3)*(4*n+3)*(14*n+25)*(n+1)/(3*n+5)/(1+2*n)/(3*n+7)/(14*n+11)
/(2+n)-8/3*(3975+20322*n+39676*n^2+37144*n^3+16736*n^4+2912*n^5)/(3*n+5)/(1+2*n
)/(3*n+7)/(14*n+11)/(2+n)*N+N^2


                      More versbosely, we have , for n>=1

      (4 n + 1) (2 n + 3) (4 n + 3) (14 n + 25) (n + 1) A[3](n)
-64/3 --------------------------------------------------------- -
          (3 n + 5) (1 + 2 n) (3 n + 7) (14 n + 11) (2 + n)

                                 2          3          4         5
        (3975 + 20322 n + 39676 n  + 37144 n  + 16736 n  + 2912 n ) A[3](n + 1)
    8/3 -----------------------------------------------------------------------
                   (3 n + 5) (1 + 2 n) (3 n + 7) (14 n + 11) (2 + n)

     + A[3](2 + n) = 0



                       subject to the initial conditions



                           A[3](1) = 1, A[3](2) = 20



                            In Maple input format:

-64/3*(4*n+1)*(2*n+3)*(4*n+3)*(14*n+25)*(n+1)/(3*n+5)/(1+2*n)/(3*n+7)/(14*n+11)
/(2+n)*A[3](n)-8/3*(3975+20322*n+39676*n^2+37144*n^3+16736*n^4+2912*n^5)/(3*n+5
)/(1+2*n)/(3*n+7)/(14*n+11)/(2+n)*A[3](n+1)+A[3](2+n) = 0


        Just for fun, the number of ways of doing it with n=, 1000, is

3148396814492324185490267589988506000043618800104357526673885470352254406512175\
6084571062922517677846098854261705840633135121837935196265184520313634514324988\
2704520845926411168890941665247349723254836348286917613920913945895139015298791\
3723083504516889544574141346575156476530518847794491749903265554453759958273057\
6069485176275605262753854648553303144676148240268928014716554260966093624930764\
9268080353516914143376592488537072887937279449169596124243542152775656705928406\
3571856375624827549886182170955771302252514345144517768731354522034458524485018\
9403990006631126845598460306192487092744550296561823241403152407142252893100454\
1906553875490237941573968114376793500244747515572082919440147946980295921423857\
1988506261197744124072088205939903035918805543294549479381426079037078190065374\
8592375034694025538195487756365498833991330767963888042002671766041106015678607\
8327116122718969553595755856507545978092867587737893874927429902588915777564590\
5183071441268840958998734764753427775831584595060060358867697919835906736127988\
1498421015285880619690416908127956356333603791666817025100307690746609517321583\
3570149233286605739848293009889626097352864209235615683397609850179660083390317\
6914886580548256294359060923242722735413462579066453275758728487844780809176991\
6900334331672463843722499934969070075405547622534890261964174703900398495561642\
0508494602006124807324806757323769200121914252657224448406906638002655718701981\
295348764091550829144266335486122472975291558313991247932041204166605392900480


                 the asymptotics of, A[3](n), to order, 6, is

  n      3/2 /     33     1105       27195       3706059        106431633
32  (1/n)    |1 - ---- + ------- - --------- + ------------ + -------------
             |    64 n         2           3              4               5
             \           8192 n    524288 n    134217728 n    8589934592 n

         23307054485   \   /      1/2
     + ----------------|  /  (8 Pi   )
                      6| /
       1099511627776 n /



                           and in Maple input format

1/8/Pi^(1/2)*32^n*(1/n)^(3/2)*(1-33/64/n+1105/8192/n^2-27195/524288/n^3+3706059
/134217728/n^4+106431633/8589934592/n^5+23307054485/1099511627776/n^6)


As a check, the ratio of the actual value at, 1000,

    to the one predicted by this formula is, 1.00000000000000000000000569064

                                    Not bad!



   -------------------------------------------------------------------------



                               Theorem number, 4

 Let , A[4](n), be the number of words,w, of length, 4 n, with exactly, 4,

    occurrences of the letter i, for i from 1 to n

that avoid the pattern 123, i.e., you can't find 1<=i1<i2<i3<=, 4 n,

    such that w[i1]<w[i2]<w[i3]

           For the sake of Sloane's OEIS, the first, 20,  terms are

[1, 70, 3199, 173860, 10203181, 631326526, 40553993125, 2678871322640, 
180830423671450, 12418980645870820, 864996624914197495, 60957211831578399100, 
4338372535640598835279, 311386494956413595138930, 22513820432313175983170649, 
1638226907374445245497453464, 119879437233268054455643664540, 
8816310717018695747221495985704, 651283473782492244839579091820690, 
48305712720059582761530066305796370]


The sequence is annihilated by the following linear recurrence operator with\
     polynomial coefficients, written in Maple input form



125*(4*n+1)*(4*n+3)*(2+n)*(n+1)*(833881955628195101*n^2+4334440398787864643*n+
5570920248415028320)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^2+
573464473150578931*n+754134476884609344)-25/2*(2+n)*(6797590926192864096514*n^5
+52313495069544026232773*n^4+146976870497476539918782*n^3+
185798998559603713597083*n^2+105551389385700444982568*n+21400378032198247784160
)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^2+573464473150578931*n+
754134476884609344)*N-5/4*(899390725561332510596160+2665788097113018431658294*n
^2+2237132112030660367540892*n+1890016298687795509670590*n^3+
797828129908309346140155*n^4+182271724749385461273198*n^5+
17187566185694281417071*n^6)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(
117658749964922315*n^2+573464473150578931*n+754134476884609344)*N^2-1/8*(
3981439903977825736232960*n+3006331443897965320930044*n^2+
1149414158136510413874416*n^3+223085148216213565033949*n^4+
18124931746162005181496*n^5+155019126659173713655*n^6+2142982572383955409843200
)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^2+573464473150578931*n+
754134476884609344)*N^3+N^4


                      More versbosely, we have , for n>=1

125 (4 n + 1) (4 n + 3) (2 + n) (n + 1)

                         2
    (833881955628195101 n  + 4334440398787864643 n + 5570920248415028320)

    A[4](n)/((2 n + 7) (4 n + 17) (n + 4) (4 n + 15) %1) - 25/2 (2 + n) (

                            5                            4
    6797590926192864096514 n  + 52313495069544026232773 n

                                 3                             2
     + 146976870497476539918782 n  + 185798998559603713597083 n

     + 105551389385700444982568 n + 21400378032198247784160) A[4](n + 1)/(

    (2 n + 7) (4 n + 17) (n + 4) (4 n + 15) %1) - 5/4 (899390725561332510596160

                                  2
     + 2665788097113018431658294 n  + 2237132112030660367540892 n

                                  3                             4
     + 1890016298687795509670590 n  + 797828129908309346140155 n

                                 5                            6
     + 182271724749385461273198 n  + 17187566185694281417071 n ) A[4](2 + n)/(

    (2 n + 7) (4 n + 17) (n + 4) (4 n + 15) %1) - 1/8 (

                                                             2
    3981439903977825736232960 n + 3006331443897965320930044 n

                                  3                             4
     + 1149414158136510413874416 n  + 223085148216213565033949 n

                                5                          6
     + 18124931746162005181496 n  + 155019126659173713655 n

     + 2142982572383955409843200) A[4](n + 3)/((2 n + 7) (4 n + 17) (n + 4)

    (4 n + 15) %1) + A[4](n + 4) = 0

                          2
%1 := 117658749964922315 n  + 573464473150578931 n + 754134476884609344



                       subject to the initial conditions



          A[4](1) = 1, A[4](2) = 70, A[4](3) = 3199, A[4](4) = 173860



                            In Maple input format:

125*(4*n+1)*(4*n+3)*(2+n)*(n+1)*(833881955628195101*n^2+4334440398787864643*n+
5570920248415028320)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^2+
573464473150578931*n+754134476884609344)*A[4](n)-25/2*(2+n)*(
6797590926192864096514*n^5+52313495069544026232773*n^4+146976870497476539918782
*n^3+185798998559603713597083*n^2+105551389385700444982568*n+
21400378032198247784160)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^
2+573464473150578931*n+754134476884609344)*A[4](n+1)-5/4*(
899390725561332510596160+2665788097113018431658294*n^2+
2237132112030660367540892*n+1890016298687795509670590*n^3+
797828129908309346140155*n^4+182271724749385461273198*n^5+
17187566185694281417071*n^6)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(
117658749964922315*n^2+573464473150578931*n+754134476884609344)*A[4](2+n)-1/8*(
3981439903977825736232960*n+3006331443897965320930044*n^2+
1149414158136510413874416*n^3+223085148216213565033949*n^4+
18124931746162005181496*n^5+155019126659173713655*n^6+2142982572383955409843200
)/(2*n+7)/(4*n+17)/(n+4)/(4*n+15)/(117658749964922315*n^2+573464473150578931*n+
754134476884609344)*A[4](n+3)+A[4](n+4) = 0


        Just for fun, the number of ways of doing it with n=, 1000, is

1492715564681093362354625290815937249763830246110807525436034559988662184209311\
2724134781383392540979020110037048475561492282289352605671972825849606397095071\
2510756123178219023165167231272387109718166181126882965169058569618011276401527\
3355018579340280512357113491805038633401811319740046042136735752942821699391672\
5218771638029994988344287689844804826680987745649682108480000033477013573658602\
6684275495821310862880948865199147931688350751816785445845184696137805342695922\
2465447196188163861409258843727430793373625114068782224030593468025972177923017\
3060659247379453777466314265610980326814115901837516912031817354291587840415849\
7956428214418920949461695696881625850681035094715093794335833747546538932002172\
4689605420979057471910664961847592806900332457345296131628889374189794565852452\
0432819130936762527667987670589306377274990700867410579279885679907388102118773\
3387185910250049036303014721067952218894922021105802667045568995001620795006733\
2589554365619907838205340729244573545020826021646479560956274418680878858458910\
8949946938726726227437057728946283036231271097728377769031144162279459674967625\
6140740215176274976609172644410305040397787259398193257026528345414884779014909\
1248502021691086122513331095183468877828807660053514470313934318931222428132456\
6414978014885131028992295353042789271728967436624551651436013812310012285416514\
0049750584187427324814008637755088233703993241229349426708087475971096064311936\
2205693466903338952531107498735147475738777263829816152691112057469203664554422\
3709456536970650147708036644426744693659162130326454814078950750907478856942341\
6689260511711110775015133918424338563519330003529085563991951970257337852237630\
6627191560030701782443973543171028995057554840229335800841780507025571141653192\
0552900313466084890051516558476308951920894554172602205802252240881885245436679\
1133129628030383652073227858849788936171465431659469052313555337149966325349728\
90


                 the asymptotics of, A[4](n), to order, 6, is

 1/2  1/2   n      3/2 /     23      1621       339199        90153721
2    3    80  (1/n)    |1 - ---- + -------- - ----------- + -------------
                       |    48 n          2             3               4
                       \           23040 n    16588800 n    3185049600 n

         115691883461       85715303805703   \   /       1/2
     + ---------------- + -------------------|  /  (36 Pi   )
                      5                     6| /
       3822059520000 n    3302259425280000 n /



                           and in Maple input format

1/36*2^(1/2)*3^(1/2)/Pi^(1/2)*80^n*(1/n)^(3/2)*(1-23/48/n+1621/23040/n^2-339199
/16588800/n^3+90153721/3185049600/n^4+115691883461/3822059520000/n^5+
85715303805703/3302259425280000/n^6)


As a check, the ratio of the actual value at, 1000,

    to the one predicted by this formula is, 0.999999999999999999999986164585

                                    Not bad!



   -------------------------------------------------------------------------



                               Theorem number, 5

 Let , A[5](n), be the number of words,w, of length, 5 n, with exactly, 5,

    occurrences of the letter i, for i from 1 to n

that avoid the pattern 123, i.e., you can't find 1<=i1<i2<i3<=, 5 n,

    such that w[i1]<w[i2]<w[i3]

           For the sake of Sloane's OEIS, the first, 20,  terms are

[1, 252, 26945, 3526452, 496632627, 73755038148, 11370526112831, 
1802608455113988, 292025514204152288, 48132038318610945704, 
8045664273096259377006, 1360731282747421987390812, 232420615523852127542961849,
40035835234057533413467195836, 6947049574459144345767787409913, 
1213187850694192250327775433235876, 213060112278129023804263616188583292, 
37605243443507557948999352634024956784, 
6667087906942820489159477434013368910307, 
1186778630445927739902666146716602793786296]


The sequence is annihilated by the following linear recurrence operator with\
     polynomial coefficients, written in Maple input form



62208/5*(6*n+5)*(1+2*n)*(6*n+1)*(2+n)*(n+1)*(
35977691436282616399086887407277643058634025605978329*n^3+
285882535463552943501778893605943560545220539172834079*n^2+
755218087425054140665508730353269605836994287699299859*n+
663122757025440444830512129264481540684096097544780704)/(5*n+17)/(5*n+18)/(5*n+
19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*n^3+
4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)-32/5*(2+n)*(
20682684343672990248675921454525852394126840880865728920473*n^7+
226076397769416093823469437214429668818022608615134944119039*n^6+
989141990028801923402652198518478623543762416699711640062534*n^5+
2215331772907630412598027898571234055990101587943071535282136*n^4+
2699753141594777779063196188413842514302630943219781222823683*n^3+
1750137631280573754613484109135599575411604050492497705294477*n^2+
522989525986658558341401546479984526706622250418715407572342*n+
40000155312544514937662720961472986570958636653479478948180)/(5*n+17)/(5*n+18)/
(5*n+19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*n^
3+4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*N-4/15*(
13546723537496159728679589343564800515837105581551686677389120+
43026467762530033758757910360252017155096355626744649145629200*n+
62093386273070321879112434836477898901595368415230194969611200*n^2+
54979652352359707014806136905612866479429876459833833415847602*n^3+
33272510943721755943228273765790569323089611199046955439849581*n^4+
13967170527470476918668627867572792035442076635439088562672390*n^5+
3859884503061997345293540231301004197399264431781078596638680*n^6+
622932367305579826963523831325425375893894893380744023095368*n^7+
43989103350760216022917286270729410533054122441680010268619*n^8)/(5*n+17)/(5*n+
18)/(5*n+19)/(n+4)/(5*n+21)/(
656486228393872890174481440882766370514517335421243*n^3+
4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*N^2-4/15*(
3740002353099313876646241205051099454225669464204109779223832*n+
4646506441120530834658707963309109108492410521664152698766862*n^2+
3200989570122994869943753872807218886042821668518908202724581*n^3+
1326770011771513987079582576031790639924795105682821537064314*n^4+
333260310210039489422524375967301246303349396525422229415308*n^5+
47834413146485695738972605784085185027192044502840653887268*n^6+
3272664364499983113393427994769622222057126079121041718719*n^7+
52860675555434422710899931418441981798739782573908924956*n^8+
1281262653546018385206479186451143604169100362060407138693120)/(5*n+17)/(5*n+18
)/(5*n+19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*
n^3+4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*N^3+N^4


                      More versbosely, we have , for n>=1

62208/5 (6 n + 5) (1 + 2 n) (6 n + 1) (2 + n) (n + 1) (

                                                           3
    35977691436282616399086887407277643058634025605978329 n

                                                               2
     + 285882535463552943501778893605943560545220539172834079 n

     + 755218087425054140665508730353269605836994287699299859 n

     + 663122757025440444830512129264481540684096097544780704) A[5](n)/(

    (5 n + 17) (5 n + 18) (5 n + 19) (n + 4) (5 n + 21) %1) - 32/5 (2 + n) (

                                                                 7
    20682684343672990248675921454525852394126840880865728920473 n

                                                                     6
     + 226076397769416093823469437214429668818022608615134944119039 n

                                                                     5
     + 989141990028801923402652198518478623543762416699711640062534 n

                                                                      4
     + 2215331772907630412598027898571234055990101587943071535282136 n

                                                                      3
     + 2699753141594777779063196188413842514302630943219781222823683 n

                                                                      2
     + 1750137631280573754613484109135599575411604050492497705294477 n

     + 522989525986658558341401546479984526706622250418715407572342 n

     + 40000155312544514937662720961472986570958636653479478948180) A[5](n + 1)

    /((5 n + 17) (5 n + 18) (5 n + 19) (n + 4) (5 n + 21) %1) - 4/15 (

    13546723537496159728679589343564800515837105581551686677389120

     + 43026467762530033758757910360252017155096355626744649145629200 n

                                                                       2
     + 62093386273070321879112434836477898901595368415230194969611200 n

                                                                       3
     + 54979652352359707014806136905612866479429876459833833415847602 n

                                                                       4
     + 33272510943721755943228273765790569323089611199046955439849581 n

                                                                       5
     + 13967170527470476918668627867572792035442076635439088562672390 n

                                                                      6
     + 3859884503061997345293540231301004197399264431781078596638680 n

                                                                     7
     + 622932367305579826963523831325425375893894893380744023095368 n

                                                                    8
     + 43989103350760216022917286270729410533054122441680010268619 n )

    A[5](2 + n)/((5 n + 17) (5 n + 18) (5 n + 19) (n + 4) (5 n + 21) %1) - 4/15

    (3740002353099313876646241205051099454225669464204109779223832 n

                                                                      2
     + 4646506441120530834658707963309109108492410521664152698766862 n

                                                                      3
     + 3200989570122994869943753872807218886042821668518908202724581 n

                                                                      4
     + 1326770011771513987079582576031790639924795105682821537064314 n

                                                                     5
     + 333260310210039489422524375967301246303349396525422229415308 n

                                                                    6
     + 47834413146485695738972605784085185027192044502840653887268 n

                                                                   7
     + 3272664364499983113393427994769622222057126079121041718719 n

                                                                 8
     + 52860675555434422710899931418441981798739782573908924956 n

     + 1281262653546018385206479186451143604169100362060407138693120)

    A[5](n + 3)/((5 n + 17) (5 n + 18) (5 n + 19) (n + 4) (5 n + 21) %1)

     + A[5](n + 4) = 0

                                                           3
%1 := 656486228393872890174481440882766370514517335421243 n

                                                             2
     + 4233692976503708702734111422895970338832268857096474 n

     + 8840701043349298040442216716805990644287291130165063 n

     + 5874384960324293748126718637456945145115359526097272



                       subject to the initial conditions



         A[5](1) = 1, A[5](2) = 252, A[5](3) = 26945, A[5](4) = 3526452



                            In Maple input format:

62208/5*(6*n+5)*(1+2*n)*(6*n+1)*(2+n)*(n+1)*(
35977691436282616399086887407277643058634025605978329*n^3+
285882535463552943501778893605943560545220539172834079*n^2+
755218087425054140665508730353269605836994287699299859*n+
663122757025440444830512129264481540684096097544780704)/(5*n+17)/(5*n+18)/(5*n+
19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*n^3+
4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*A[5](n)-32/5*(2+n)*(
20682684343672990248675921454525852394126840880865728920473*n^7+
226076397769416093823469437214429668818022608615134944119039*n^6+
989141990028801923402652198518478623543762416699711640062534*n^5+
2215331772907630412598027898571234055990101587943071535282136*n^4+
2699753141594777779063196188413842514302630943219781222823683*n^3+
1750137631280573754613484109135599575411604050492497705294477*n^2+
522989525986658558341401546479984526706622250418715407572342*n+
40000155312544514937662720961472986570958636653479478948180)/(5*n+17)/(5*n+18)/
(5*n+19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*n^
3+4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*A[5](n+1)-4/15*(
13546723537496159728679589343564800515837105581551686677389120+
43026467762530033758757910360252017155096355626744649145629200*n+
62093386273070321879112434836477898901595368415230194969611200*n^2+
54979652352359707014806136905612866479429876459833833415847602*n^3+
33272510943721755943228273765790569323089611199046955439849581*n^4+
13967170527470476918668627867572792035442076635439088562672390*n^5+
3859884503061997345293540231301004197399264431781078596638680*n^6+
622932367305579826963523831325425375893894893380744023095368*n^7+
43989103350760216022917286270729410533054122441680010268619*n^8)/(5*n+17)/(5*n+
18)/(5*n+19)/(n+4)/(5*n+21)/(
656486228393872890174481440882766370514517335421243*n^3+
4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*A[5](2+n)-4/15*(
3740002353099313876646241205051099454225669464204109779223832*n+
4646506441120530834658707963309109108492410521664152698766862*n^2+
3200989570122994869943753872807218886042821668518908202724581*n^3+
1326770011771513987079582576031790639924795105682821537064314*n^4+
333260310210039489422524375967301246303349396525422229415308*n^5+
47834413146485695738972605784085185027192044502840653887268*n^6+
3272664364499983113393427994769622222057126079121041718719*n^7+
52860675555434422710899931418441981798739782573908924956*n^8+
1281262653546018385206479186451143604169100362060407138693120)/(5*n+17)/(5*n+18
)/(5*n+19)/(n+4)/(5*n+21)/(656486228393872890174481440882766370514517335421243*
n^3+4233692976503708702734111422895970338832268857096474*n^2+
8840701043349298040442216716805990644287291130165063*n+
5874384960324293748126718637456945145115359526097272)*A[5](n+3)+A[5](n+4) = 0


        Just for fun, the number of ways of doing it with n=, 1000, is

1483272866638701923151738439852443066155558520312446850795108676973371060613332\
2562635619143794191725373540429731931292182612362554854562449578156077661669074\
8533151899895683543875423242586067357896053033712191410263670877038296875727649\
9363947710432581950499478262843952974397010318061319916356972436289203691691671\
4354477460328426491341208508345194438620807891010224093386321541333356859916706\
3809262623947977615726179206964433145245366698011761737304934573557734682787311\
6090094460847937277025705900471197856066120621090971396345647894667449986290776\
6836582519783858482358758895049306496039968208022639908745806012712821199052545\
0953978067484753048200864197679177566145538638739352893698193913549478199425779\
8196042545316708068226839807773653361704268082339111067391878758574625502895128\
1091359234326784947956023706554154307677576030703999978099265575111100937917005\
6253984156719695083006340838372734539908314560970167128684579800074824604200914\
1559008104265909048243660872456342811590027383023942733178530093251422641873828\
2703054808162519936115839223026270464833749012748736662011372070710853706097661\
4489147823807338581069917659820917915411245255153894865320187847745170057582994\
1404748194380418345070460846736893263293217819850449478826927344838189783014151\
5547270670390755305220379699342420005790455694307552352397241000789339246712030\
5019650614469992248658478380193172918470496925962216066677300416148592731252004\
8502595150712674805671015972074452763761123812508912478242860942155595747847715\
8554394666813424544977728800966895912444702073127351386866887235314596755562688\
0763976000523937721317577718693679201039017946184062337158907721123022047703273\
2029412954816785341084391047341422416664010548599831001150494231543918833454741\
1552852569471262986447143665240035639065239741687895871736975284001840251817248\
2550569486956737506715780928330841620950455390698023141105124242771094073710368\
0815910014908672840425329901406927176741694555886054682496959435297162933785679\
3116733572745309994597543716777209159255865278769362313248488581108622367404987\
0745220774037800557678556113275492010744758469966114783652305581265957945893914\
0849364154283974547924489651235127316318679304263569964632200242984965991520147\
877187003093031684558354103573198621903976221740926839882907991080


                 the asymptotics of, A[5](n), to order, 6, is

   1/2    n      3/2 /     471       389141        162387477
3 3    192  (1/n)    |1 - ------ + ----------- - --------------
                     |    1000 n             2                3
                     \             10000000 n    50000000000 n

         7335332569299        207687570285951063       46925230488922877377   \
     + ------------------ + ---------------------- + -------------------------|
                        4                        5                           6|
       200000000000000 n    5000000000000000000 n    2000000000000000000000 n /

       /        1/2
      /  (125 Pi   )
     /



                           and in Maple input format

3/125*3^(1/2)/Pi^(1/2)*192^n*(1/n)^(3/2)*(1-471/1000/n+389141/10000000/n^2-\
162387477/50000000000/n^3+7335332569299/200000000000000/n^4+207687570285951063/
5000000000000000000/n^5+46925230488922877377/2000000000000000000000/n^6)


As a check, the ratio of the actual value at, 1000,

    to the one predicted by this formula is, 0.999999999999999999999952455399

                                    Not bad!

         This concludes this article, that took, 754.794, to generate.