Proof of the identity at the bottom of page 7 of the Zeilberger-Zudilin paper by Shalosh B. Ekhad Proposition Let F1 be n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n) Let F2 be (n + m) k (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k) The sum of F1 with resepect to n0 equals the sum of F2 with respect to k We will prove that both sums satisfy the same second-order linear recurrence\ s with respect to each of the three discrete variables m,n,j ----------------------------- For the variable m, and F1, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(m, n0), be given by n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n) and let, SUM(m), be the sum of, F(m, n0), with respect to, n0 SUM(m), satisfies the following linear recurrence equation 2 (2 m + 2 n + 1) (m + n + 1) SUM(m) + 2 2 2 (-j - 4 j m - 4 j n - 8 m + 8 m n + 16 n - 5 j - 16 m + 14 n - 8) SUM(m + 1) + (j + 2 m - 4 n + 3) (j + 2 m - 4 n + 2) SUM(m + 2) =0. PROOF: We cleverly construct, G(m, n0), := - 4 (n + m + 1/2) (-6 n + j + n0 - 1) (m + n + 1) 2 n0 (-12 n + (-12 m + 4 n0 - 22) n + (n0 - 1) (-n0 + j + 4 m + 6)) n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n)/((2 n + 2 m - n0 + 1) (2 n + 2 m + 2 - n0) (2 n + 2 m + 3 - n0) (2 n + 2 m + 4 - n0)) with the motive that 2 (2 m + 2 n + 1) (m + n + 1) F(m, n0) + 2 2 2 (-j - 4 j m - 4 j n - 8 m + 8 m n + 16 n - 5 j - 16 m + 14 n - 8) F(m + 1, n0) + (j + 2 m - 4 n + 3) (j + 2 m - 4 n + 2) F(m + 2, n0) = G(m, n0 + 1) - G(m, n0), (Check!) and the theorem follows upon summing with respect to, n0, . QED. For the variable m, and F2, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(m, k), be given by (n + m) k (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k) and let, SUM(m), be the sum of, F(m, k), with respect to, k SUM(m), satisfies the following linear recurrence equation 2 (2 m + 2 n + 1) (m + n + 1) SUM(m) + 2 2 2 (-j - 4 j m - 4 j n - 8 m + 8 m n + 16 n - 5 j - 16 m + 14 n - 8) SUM(m + 1) + (j + 2 m - 4 n + 3) (j + 2 m - 4 n + 2) SUM(m + 2) =0. PROOF: We cleverly construct, G(m, k), := 4 3 4 (n + m + 1/2) (m + n + 1) (4 n + (-5 j - 14 m - 23) n 2 2 3 + (6 m + (9 j - 6) m + 2 (j + k/2 - 2) (j + 2 k + 6)) n + (16 m 2 2 2 2 + (6 j + 66) m + (j + (-8 k + 28) j - 8 k + 8 k + 78) m + (-3 k + 4) j 2 2 4 3 + (-2 k - 6 k + 25) j - 2 k + 2 k + 25) n - 8 m + (-8 j - 32) m 2 2 2 + (-j + (-4 k - 25) j + 8 k - 8 k - 42) m 2 2 2 2 2 + (-2 j + (4 k - 18 k - 20) j + 16 k - 16 k - 20) m + (k - 2 k) j 2 2 (n + m) + (3 k - 14 k - 2) j + 10 k - 10 k - 2) (-2 n + m + j - k) (-1) k (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k)/( (-n - m - 1 + k) (-4 n + 2 m + j) (-4 n + 2 m + 1 + j) (-n - m - 2 + k) (n + m + 1 + k)) with the motive that 2 (2 m + 2 n + 1) (m + n + 1) F(m, k) + 2 2 2 (-j - 4 j m - 4 j n - 8 m + 8 m n + 16 n - 5 j - 16 m + 14 n - 8) F(m + 1, k) + (j + 2 m - 4 n + 3) (j + 2 m - 4 n + 2) F(m + 2, k) = G(m, k + 1) - G(m, k), (Check!) and the theorem follows upon summing with respect to, k, . QED. One can see that these are recurrences are identical -------------------------------------------- For the variable n, and F1, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(n, n0), be given by n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n) and let, SUM(n), be the sum of, F(n, n0), with respect to, n0 SUM(n), satisfies the following linear recurrence equation -128 (2 m + 2 n + 1) (m + n + 1) (j + 2 m - 4 n - 2) (j + 2 m - 4 n - 4) 4 3 3 2 2 (j + 2 m - 4 n - 1) (j + 2 m - 4 n - 3) (j + 8 j m - 16 j n + 16 j m 2 2 2 2 2 3 2 2 - 112 j m n + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n 3 3 2 2 2 - 864 m n - 26 j - 184 j m + 284 j n - 240 j m + 1896 j m n 2 2 2 3 2 - 672 j n + 1440 m n - 4248 m n - 72 n + 227 j + 1552 j m - 1028 j n 2 2 + 1184 m - 6944 m n - 388 n - 514 j - 3776 m - 692 n - 408) SUM(n) + ( 10 9 9 8 2 8 8 2 7 3 j + 20 j m - 28 j n + 160 j m - 568 j m n + 280 j n + 640 j m 7 2 7 2 7 3 6 4 6 3 - 4416 j m n + 6528 j m n - 512 j n + 1280 j m - 16384 j m n 6 2 2 6 3 6 4 5 5 + 52032 j m n - 35200 j m n - 14176 j n + 1024 j m 5 4 5 3 2 5 2 3 5 4 - 28672 j m n + 181504 j m n - 337664 j m n + 39296 j m n 5 5 4 5 4 4 2 4 3 3 + 152704 j n - 18432 j m n + 274176 j m n - 1138176 j m n 4 2 4 4 5 4 6 3 5 2 + 1287936 j m n + 647424 j m n - 764928 j n + 138240 j m n 3 4 3 3 3 4 3 2 5 3 6 - 1451520 j m n + 4423680 j m n - 2764800 j m n - 4396032 j m n 3 7 2 5 3 2 4 4 2 3 5 + 2142720 j n - 552960 j m n + 4596480 j m n - 10920960 j m n 2 2 6 2 7 2 8 5 4 + 2059776 j m n + 13243392 j m n - 3207168 j n + 1244160 j m n 4 5 3 6 2 7 8 - 8709120 j m n + 15427584 j m n + 3483648 j m n - 19906560 j m n 9 5 5 4 6 3 7 + 1990656 j n - 1492992 m n + 7464960 m n - 8957952 m n 2 8 9 9 8 8 7 2 - 5971968 m n + 11943936 m n - 31 j - 636 j m + 606 j n - 4992 j m 7 7 2 6 3 6 2 + 14496 j m n - 1248 j n - 18688 j m + 117312 j m n 6 2 6 3 5 4 5 3 - 114528 j m n - 69088 j n - 33024 j m + 413952 j m n 5 2 2 5 3 5 4 4 5 - 1136448 j m n + 127104 j m n + 894048 j n - 21504 j m 4 4 4 3 2 4 2 3 4 4 + 632064 j m n - 3892224 j m n + 5725824 j m n + 3991296 j m n 4 5 3 5 3 4 2 3 3 3 - 5300544 j n + 322560 j m n - 5022720 j m n + 20160000 j m n 3 2 4 3 5 3 6 2 5 2 - 15022080 j m n - 31233024 j m n + 17132544 j n - 1935360 j m n 2 4 3 2 3 4 2 2 5 + 21219840 j m n - 62173440 j m n + 11688192 j m n 2 6 2 7 5 3 4 4 + 108497664 j m n - 28829952 j n + 5806080 j m n - 50284800 j m n 3 5 2 6 7 8 + 105090048 j m n + 32306688 j m n - 185213952 j m n + 19243008 j n 5 4 4 5 3 6 2 7 - 8709120 m n + 51632640 m n - 70668288 m n - 57977856 m n 8 9 8 7 7 6 2 + 124416000 m n + 995328 n + 302 j + 7640 j m - 304 j n + 63616 j m 6 6 2 5 3 5 2 - 115024 j m n - 129752 j n + 228608 j m - 1221312 j m n 5 2 5 3 4 4 4 3 + 52800 j m n + 2095424 j n + 353024 j m - 4294144 j m n 4 2 2 4 3 4 4 3 5 + 9070464 j m n + 10057856 j m n - 15181216 j n + 179200 j m 3 4 3 3 2 3 2 3 - 5608960 j m n + 33326080 j m n - 30412160 j m n 3 4 3 5 2 5 2 4 2 - 92325760 j m n + 57977728 j n - 2150400 j m n + 35535360 j m n 2 3 3 2 2 4 2 5 - 136888320 j m n + 21410880 j m n + 377765760 j m n 2 6 5 2 4 3 - 111498816 j n + 9676800 j m n - 112250880 j m n 3 4 2 5 6 7 + 288645120 j m n + 131272704 j m n - 745445376 j m n + 80289792 j n 5 3 4 4 3 5 2 6 - 19353600 m n + 144184320 m n - 231026688 m n - 246786048 m n 7 8 7 6 6 + 568442880 m n + 10838016 n + 498 j - 34616 j m - 109964 j n 5 2 5 5 2 4 3 - 416064 j m - 137472 j m n + 2451336 j n - 1523200 j m 4 2 4 2 4 3 3 4 + 6000000 j m n + 12811200 j m n - 22971280 j n - 2016000 j m 3 3 3 2 2 3 3 + 23600640 j m n - 27944640 j m n - 145002240 j m n 3 4 2 5 2 4 2 3 2 + 107508000 j n - 752640 j m + 25509120 j m n - 145178880 j m n 2 2 3 2 4 2 5 5 + 7403520 j m n + 723716160 j m n - 241888128 j n + 6773760 j m n 4 2 3 3 2 4 - 120718080 j m n + 407877120 j m n + 297414720 j m n 5 6 5 2 4 3 - 1693471104 j m n + 188547264 j n - 20321280 m n + 207567360 m n 3 4 2 5 6 7 - 404248320 m n - 599446656 m n + 1493655552 m n + 51767424 n 6 5 5 4 2 4 - 34935 j - 100812 j m + 1424508 j n + 1372000 j m + 8140040 j m n 4 2 3 3 3 2 3 2 - 19313312 j n + 6016640 j m - 11001984 j m n - 127103616 j m n 3 3 2 4 2 3 2 2 2 + 117692192 j n + 6594560 j m - 73862656 j m n - 19638528 j m n 2 3 2 4 5 4 + 822080768 j m n - 320989936 j n + 1662976 j m - 62261248 j m n 3 2 2 3 4 + 311594752 j m n + 398814976 j m n - 2371328896 j m n 5 5 4 2 3 3 + 271485568 j n - 9977856 m n + 161839104 m n - 407404032 m n 2 4 5 6 5 - 903775488 m n + 2484084480 m n + 142528512 n + 326769 j 4 4 3 2 3 + 2037700 j m - 8517482 j n - 1232448 j m - 58646112 j m n 3 2 2 3 2 2 2 2 + 75779064 j n - 14355712 j m - 21693504 j m n + 551904864 j m n 2 3 4 3 2 2 - 265558960 j n - 12235776 j m + 121524480 j m n + 312177600 j m n 3 4 5 4 - 2091247872 j m n + 243207504 j n - 1806336 m + 64382976 m n 3 2 2 3 4 5 - 235807488 m n - 860162688 m n + 2706790464 m n + 249509088 n 4 3 3 2 2 - 1530376 j - 11059040 j m + 26430256 j n - 6435456 j m 2 2 2 3 2 + 201932544 j m n - 132823152 j n + 18821632 j m + 130679808 j m n 2 3 4 3 - 1131148224 j m n + 129333184 j n + 10100736 m - 72526848 m n 2 2 3 4 3 - 500829696 m n + 1928409984 m n + 288102720 n + 3824572 j 2 2 2 + 30912496 j m - 36341768 j n + 22366464 j m - 341995584 j m n 2 3 2 2 + 34399440 j n - 9169920 m - 161708544 m n + 864278208 m n 3 2 2 + 219293856 n - 4099376 j - 44097728 j m + 1482112 j n - 21934080 m 2 + 220718208 m n + 105912768 n - 816768 j + 24454656 m + 29355264 n 4 + 3538944) SUM(n + 1) + 3 (j - 3 n - 6) (n + 2) (3 n + 4) (3 n + 5) (j 3 3 2 2 2 2 2 2 + 8 j m - 16 j n + 16 j m - 112 j m n + 88 j n - 144 j m n 2 3 2 2 3 3 2 + 576 j m n - 144 j n + 432 m n - 864 m n - 10 j - 72 j m 2 2 2 2 2 3 + 108 j n - 96 j m + 744 j m n - 240 j n + 576 m n - 1656 m n - 72 n 2 2 2 + 31 j + 232 j m - 116 j n + 176 m - 1040 m n - 172 n - 14 j - 216 m - 132 n - 32) (j - 3 n - 4) (j - 3 n - 5) SUM(n + 2) =0. PROOF: We cleverly construct, G(n, n0), := -4/3 (m + n + 1/2) (-6 n + j + n0 - 1) (m + n + 1) ( 18 (-3718911688704 j - 22313470132224 m - 1859455844352) n + ( 2 12189766090752 j + (70659322085376 m + 7157905293312 n0 - 72518777929728) j 2 - 11156735066112 m + (42947431759872 n0 - 472301784465408) m 17 3 + 3578952646656 n0 - 40081603756032) n + (-18603444731904 j 2 + (-102909884203008 m - 22165735587840 n0 + 233943760355328) j + ( 2 43067396653056 m + (-129182195810304 n0 + 1415265833189376) m 2 - 6263435870208 n0 + 131035255676928 n0 - 651685400322048) j 3 2 + 15475471220736 m + (15715401007104 n0 - 202860634374144) m 2 + (-37580615221248 n0 + 853421865467904 n0 - 4665294736883712) m 2 16 - 3131717935104 n0 + 72550292004864 n0 - 403599112003584) n + ( 4 17558843019264 j 3 + (91219971833856 m + 31757594075136 n0 - 340502242295808) j + ( 2 -75058034835456 m + (177270703718400 n0 - 1945083366752256) m 2 2 + 18251283750912 n0 - 398157269630976 n0 + 2079862025158656) j + ( 3 2 -45946554089472 m + (-63550112219136 n0 + 748789582086144) m 2 + (107225395494912 n0 - 2420651775909888 n0 + 13183288730984448) m 3 2 + 3274023960576 n0 - 107050357112832 n0 + 1101573030260736 n0 4 - 3570764015517696) j + 719789359104 m 3 + (-30668444467200 n0 + 289432719065088) m 2 2 + (-7430406193152 n0 + 261759204605952 n0 - 1696616723791872) m + ( 3 2 19644143763456 n0 - 697153241456640 n0 + 7888778451836928 n0 3 2 - 28563285848481792) m + 1637011980288 n0 - 59392881967104 n0 15 5 + 684773071110144 n0 - 2521704957345792) n + (-11480276385792 j 4 + (-54791501524992 m - 27937860636672 n0 + 303291478517760) j + ( 2 79125661679616 m + (-146566033293312 n0 + 1622358938640384) m 2 3 - 24432685682688 n0 + 542097571921920 n0 - 2881167875555328) j + ( 3 2 62733899464704 m + (111746008055808 n0 - 1236631979999232) m 2 + (-138376619827200 n0 + 3123048590401536 n0 - 17031652805246976) m 3 2 - 8931619602432 n0 + 305029951248384 n0 - 3300756761180160 n0 2 4 + 11363344262000640) j + (-4612843634688 m 3 + (88010373464064 n0 - 807826821414912) m 2 2 + (36402895355904 n0 - 1018883121905664 n0 + 5983515313717248) m + ( 3 2 -53068914229248 n0 + 1867870156161024 n0 - 21018736624730112 n0 4 3 + 75821518907633664) m - 1123082330112 n0 + 51906207891456 n0 2 - 836918921631744 n0 + 5623544749860864 n0 - 13284992893261824) j 5 4 - 3088128540672 m + (2863678095360 n0 + 5553213603840) m 2 3 + (25954340143104 n0 - 528182850650112 n0 + 2501537298923520) m + ( 3 2 -149203648512 n0 - 108719635636224 n0 + 1986714140811264 n0 2 4 3 - 8642931356860416) m + (-6738493980672 n0 + 337946837188608 n0 2 - 5994860542009344 n0 + 45035633853333504 n0 - 121417909197484032) m 4 3 2 - 561541165056 n0 + 28868097171456 n0 - 522521396619264 n0 14 6 + 3995828909144064 n0 - 10951704878966784) n + (5517770637312 j 5 + (23423449964544 m + 16882814665728 n0 - 185848636916736) j + ( 2 -56830698676224 m + (80982510759936 n0 - 914372720418816) m 2 4 + 19920994993152 n0 - 448261606628352 n0 + 2413029137292288) j + ( 3 2 -52182793617408 m + (-116263825735680 n0 + 1226582232809472) m 2 + (106464641421312 n0 - 2420525029847040 n0 + 13308731408830464) m 3 2 + 11105841623040 n0 - 386428793278464 n0 + 4259935271024640 n0 3 4 - 14949418007052288) j + (9783266033664 m 3 + (-115901136175104 n0 + 1034224653828096) m 2 2 + (-68251809779712 n0 + 1697169975865344 n0 - 9337404119298048) m + ( 3 2 64251257094144 n0 - 2255914689269760 n0 + 25383598540259328 n0 4 3 - 91672437604184064) m + 2849681461248 n0 - 137923395895296 n0 2 2 + 2342998359032832 n0 - 16744525272299520 n0 + 42672596475635712) j + ( 5 4 8989627318272 m + (-3471887204352 n0 - 53069308379136) m 2 3 + (-71907343958016 n0 + 1416521594732544 n0 - 6531080986755072) m + ( 3 2 -7985584226304 n0 + 531786715361280 n0 - 7461107244564480 n0 2 4 3 + 29117752405254144) m + (17184893313024 n0 - 853027709607936 n0 2 + 15017496907124736 n0 - 112220029929793536 n0 + 301426964083574784) m 5 4 3 + 259695009792 n0 - 16400617648128 n0 + 374778246214656 n0 2 - 3955257077483520 n0 + 19399961314704384 n0 - 35291995670808576) j 6 5 + 754617876480 m + (4752157704192 n0 - 46099412582400) m 2 4 + (-6160132767744 n0 + 54277019467776 n0 - 7030200729600) m + ( 3 2 -11842125889536 n0 + 409890855002112 n0 - 4190192320389120 n0 3 4 3 + 13256714017824768) m + (1643909603328 n0 - 8168720596992 n0 2 2 - 705202471931904 n0 + 9070282863783936 n0 - 29918992534142976) m + ( 5 4 3 1558170058752 n0 - 106722167611392 n0 + 2684444883517440 n0 2 - 31708851161960448 n0 + 177770314064388096 n0 - 380377142046978048) m 5 4 3 + 129847504896 n0 - 9146704674816 n0 + 235168404129792 n0 2 13 - 2830293545269248 n0 + 16140646050121728 n0 - 35095303461113856) n + ( 7 -2018388105216 j 6 + (-7235537891328 m - 7424177734656 n0 + 83202652293120) j + ( 2 29623051468800 m + (-31157011169280 n0 + 365883679604736) m 2 5 - 11050806979584 n0 + 252767880142848 n0 - 1379420520744960) j + ( 3 2 29487768477696 m + (81228737986560 n0 - 823604014374912) m 2 + (-53952353427456 n0 + 1250504058433536 n0 - 7003120116851712) m 3 2 - 8334257107968 n0 + 294759951284736 n0 - 3296922282358272 n0 4 4 + 11723504116271616) j + (-11247246213120 m 3 + (92989525671936 n0 - 804701893607424) m 2 2 + (71799060234240 n0 - 1655338077868032 n0 + 8676701557899264) m + ( 3 2 -45868370577408 n0 + 1618891058626560 n0 - 18349194028916736 n0 4 3 + 66771140799817728) m - 3266655353856 n0 + 161511296039424 n0 2 3 - 2800382510960640 n0 + 20428397541711360 n0 - 53193491700286464) j + ( 5 4 -12000681639936 m + (-1154500853760 n0 + 115745966530560) m 2 3 + (90885974593536 n0 - 1736063396868096 n0 + 7794429637570560) m + ( 3 2 20357826785280 n0 - 949767720763392 n0 + 11745991179141120 n0 2 4 3 - 42805223201396736) m + (-19439327655936 n0 + 960982896015360 n0 2 - 16889322375078912 n0 + 126215803443299328 n0 - 339335230586379264) m 5 4 3 - 608007997440 n0 + 40354071657984 n0 - 974326888539648 n0 2 2 + 10960291576470528 n0 - 58099336734348288 n0 + 116692482328856064) j + 6 5 (-1710467186688 m + (-13228386877440 n0 + 124560645046272) m 2 4 + (12460869550080 n0 - 76590167113728 n0 - 244912145522688) m + ( 3 2 31851234533376 n0 - 1055136356757504 n0 + 10444350466584576 n0 3 4 3 - 32194570877595648) m + (-1130532360192 n0 - 102672706000896 n0 2 2 + 3516009205764096 n0 - 33019642301552640 n0 + 96367467601336320) m + ( 5 4 3 -3733121986560 n0 + 252808175609856 n0 - 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20997120 n0 + 249008640 n0 2 3 5 - 3292695552 n0 + 18607988736 n0 - 34398203904) m + (-124800 n0 4 3 2 - 16030080 n0 + 524233344 n0 - 4195850880 n0 + 13142783232 n0 2 6 - 14446949376) m + ((-3072 n0 + 39936) m 2 5 + (-1536 n0 + 107520 n0 - 993792) m 3 2 4 + (2048 n0 - 48384 n0 - 257024 n0 + 5276160) m 4 3 2 3 + (432 n0 - 52512 n0 + 1262544 n0 - 9291744 n0 + 16745808) m + 5 4 3 2 2 (-4 n0 - 4884 n0 + 329700 n0 - 6442404 n0 + 47474872 n0 - 115904616) m + 5 4 3 2 (-14 n0 + 9942 n0 - 366482 n0 + 4072146 n0 - 11059328 n0 - 24132864) m 5 4 3 2 - 12 n0 + 3006 n0 - 277365 n0 + 6381360 n0 - 55769062 n0 + 167084965) 7 2 6 j + ((-3072 n0 + 101376 n0 - 675840) m 3 2 5 + (1024 n0 + 37632 n0 - 1622272 n0 + 10466304) m 4 3 2 4 + (768 n0 - 69056 n0 + 1026816 n0 + 75200 n0 - 33856896) m + ( 5 4 3 2 -32 n0 - 13584 n0 + 896640 n0 - 16247184 n0 + 104651936 n0 - 210486432) 3 m + ( 5 4 3 2 -12 n0 + 74196 n0 - 3384316 n0 + 53923536 n0 - 347116772 n0 + 787463076 2 ) m + 5 4 3 2 (254 n0 - 55998 n0 + 762578 n0 + 4384080 n0 - 131973330 n0 + 568161686) 5 4 3 2 m + 300 n0 - 70995 n0 + 3471945 n0 - 55938762 n0 + 380150680 n0 6 7 - 939200374) j + (-1007616 m 3 2 6 + (21504 n0 - 924672 n0 + 12773376 n0 - 29604864) m 4 3 2 5 + (-16128 n0 + 404992 n0 + 499200 n0 - 91504384 n0 + 332112384) m + ( 5 4 3 2 1344 n0 + 194880 n0 - 7465856 n0 + 92791104 n0 - 190068928 n0 4 5 4 3 2 - 364946048) m + (-2912 n0 - 1416048 n0 + 50421696 n0 - 735465600 n0 3 5 4 + 4000173392 n0 - 7671110592) m + (-17268 n0 + 2923332 n0 3 2 2 - 70113172 n0 + 956929444 n0 - 5509363328 n0 + 11401050160) m + ( 5 4 3 2 -4102 n0 + 2754390 n0 - 130240126 n0 + 1733089802 n0 - 10035893852 n0 5 4 3 2 + 21966114728) m + 16068 n0 - 2742772 n0 + 85779112 n0 - 972080552 n0 4 5 4 + 4876510576 n0 - 9252682664) j + (-1280 m + (-960 n0 + 27328) m 2 3 + (1120 n0 - 13536 n0 - 52016) m 3 2 2 + (420 n0 - 21180 n0 + 251456 n0 - 703584) m 4 3 2 4 + (20 n0 - 2558 n0 + 66738 n0 - 528964 n0 + 973012) m - 31 n0 3 2 9 13 + 1408 n0 - 4793 n0 - 239673 n0 + 1708793) j + (n0 - 1) j 2 2 12 3 + (-12 m + (20 n0 - 74) m + 3 n0 - 55 n0 - 2) j + (-144 m 2 2 3 2 + (112 n0 - 40) m + (60 n0 - 1308 n0 + 3864) m + 3 n0 - 138 n0 11 3 2 6 + 1060 n0 + 2351) j + ((-1024 n0 + 82944 n0 - 1421312 n0 + 5953536) m 4 3 2 5 + (768 n0 - 32256 n0 - 357888 n0 + 14749440 n0 - 70415616) m + 5 4 3 2 (-64 n0 - 19008 n0 + 983424 n0 - 12257472 n0 + 18131968 n0 + 139273152) 4 5 4 3 2 m + (544 n0 + 184944 n0 - 8633792 n0 + 134403120 n0 - 781316864 n0 3 5 4 3 2 + 1560164400) m + (1476 n0 - 611820 n0 + 20515032 n0 - 290553840 n0 2 5 4 3 + 1714571912 n0 - 3661272176) m + (-1266 n0 - 62970 n0 + 12604728 n0 2 5 4 - 232581816 n0 + 1722166800 n0 - 4581622812) m - 3060 n0 + 603717 n0 3 2 5 - 22634982 n0 + 296975967 n0 - 1699391998 n0 + 3630026644) j + ( 9 8 5898240 m + (-10321920 n0 - 97812480) m 2 7 + (6266880 n0 + 185917440 n0 + 105332736) m 3 2 6 + (-3045376 n0 - 86913024 n0 - 694224896 n0 + 1387843584) m + 4 3 2 5 (1270272 n0 + 9993216 n0 + 488363520 n0 - 177361920 n0 - 3228340224) m 5 4 3 2 + (-98176 n0 - 6609792 n0 + 22819456 n0 - 1631958144 n0 4 5 4 + 7060404480 n0 - 5870748672) m + (-168320 n0 + 31191552 n0 3 2 3 - 448911488 n0 + 6418135296 n0 - 35582442752 n0 + 65010948096) m + ( 5 4 3 2 321760 n0 + 4775904 n0 - 503760672 n0 + 3759594144 n0 - 8287747648 n0 2 5 4 3 + 2148209664) m + (535520 n0 - 77849184 n0 + 1960183520 n0 2 5 - 19016089248 n0 + 83417682752 n0 - 140339599104) m + 8256 n0 4 3 2 - 1580352 n0 + 56084544 n0 - 673132224 n0 + 3457521024 n0 - 6577712640) 8 j + (50921472 n0 + 100073472) m 2 7 + (-30916608 n0 - 247873536 n0 + 205062144) m 3 2 6 + (8626176 n0 + 164290560 n0 + 195084288 n0 - 1133297664) m + 4 3 2 5 (-1468416 n0 - 35008512 n0 - 350423040 n0 + 983261184 n0 + 815972352) m 5 4 3 2 + (84480 n0 + 5884416 n0 + 26774016 n0 + 702463488 n0 - 4270033920 n0 4 6 5 + 4785180672) m + (-1024 m + (-3072 n0 + 53760) m 2 4 + (896 n0 + 27648 n0 - 506496) m 3 2 3 + (1344 n0 - 57008 n0 + 501744 n0 - 415360) m 4 3 2 2 + (136 n0 - 17872 n0 + 486456 n0 - 4337984 n0 + 11431708) m 4 3 2 4 + (-720 n0 + 44424 n0 - 756340 n0 + 4236624 n0 - 4652366) m + 230 n0 3 2 8 9 + 3154 n0 - 362213 n0 + 5138873 n0 - 20568222) j + (-393216 m 8 2 7 + (688128 n0 + 16416768) m + (-417792 n0 - 29712384 n0 - 111894528) m 3 2 6 + (844800 n0 + 129024 n0 + 384168960 n0 - 481845248) m + 4 3 2 5 (-566016 n0 + 5295616 n0 - 178720512 n0 - 667012096 n0 + 2926202880) m 5 4 3 2 + (46656 n0 + 3441216 n0 - 58113344 n0 + 1259594176 n0 - 4175265280 n0 4 5 4 3 + 1889303808) m + (47104 n0 - 19143168 n0 + 374161920 n0 2 3 5 - 5470518976 n0 + 29467745216 n0 - 53632705280) m + (-249600 n0 4 3 2 + 9066656 n0 + 80734176 n0 + 134017920 n0 - 6817078848 n0 + 20285217344 2 5 4 3 2 ) m + (-288712 n0 + 51364872 n0 - 1484633128 n0 + 15491281624 n0 5 4 - 72252081904 n0 + 129107834240) m + 51312 n0 - 7077248 n0 3 2 2 + 161210000 n0 - 1403656576 n0 + 5437343488 n0 - 7916999808) j 2 3 4 5 9 + 1321731072 n0 - 144516096 n0 + 6257664 n0 - 46080 n0 - 29097984 m n0 - 5506025472 n0) (n0 - 3 + j)) n0 (-2) binomial(2 n + 2 m, n0) / binomial(6 n - j - n0, 3 n) / ((-3 n + j + n0 - 1) (2 n + 2 m - n0 + 1) / (2 n + 2 m + 3 - n0) (2 n + 2 m + 2 - n0) (-3 n + j + n0 - 2) (-3 n - 3 + j + n0) (-3 n - 4 + j + n0) (2 n + 2 m + 4 - n0) 3 2 (9 n + 18 n + 11 n + 2) (-3 n - 5 + j + n0) (-3 n - 6 + j + n0)) with the motive that -128 (2 m + 2 n + 1) (m + n + 1) (j + 2 m - 4 n - 2) (j + 2 m - 4 n - 4) 4 3 3 2 2 (j + 2 m - 4 n - 1) (j + 2 m - 4 n - 3) (j + 8 j m - 16 j n + 16 j m 2 2 2 2 2 3 2 2 - 112 j m n + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n 3 3 2 2 2 - 864 m n - 26 j - 184 j m + 284 j n - 240 j m + 1896 j m n 2 2 2 3 2 - 672 j n + 1440 m n - 4248 m n - 72 n + 227 j + 1552 j m - 1028 j n 2 2 + 1184 m - 6944 m n - 388 n - 514 j - 3776 m - 692 n - 408) F(n, n0) + ( 10 9 9 8 2 8 8 2 7 3 j + 20 j m - 28 j n + 160 j m - 568 j m n + 280 j n + 640 j m 7 2 7 2 7 3 6 4 6 3 - 4416 j m n + 6528 j m n - 512 j n + 1280 j m - 16384 j m n 6 2 2 6 3 6 4 5 5 + 52032 j m n - 35200 j m n - 14176 j n + 1024 j m 5 4 5 3 2 5 2 3 5 4 - 28672 j m n + 181504 j m n - 337664 j m n + 39296 j m n 5 5 4 5 4 4 2 4 3 3 + 152704 j n - 18432 j m n + 274176 j m n - 1138176 j m n 4 2 4 4 5 4 6 3 5 2 + 1287936 j m n + 647424 j m n - 764928 j n + 138240 j m n 3 4 3 3 3 4 3 2 5 3 6 - 1451520 j m n + 4423680 j m n - 2764800 j m n - 4396032 j m n 3 7 2 5 3 2 4 4 2 3 5 + 2142720 j n - 552960 j m n + 4596480 j m n - 10920960 j m n 2 2 6 2 7 2 8 5 4 + 2059776 j m n + 13243392 j m n - 3207168 j n + 1244160 j m n 4 5 3 6 2 7 8 - 8709120 j m n + 15427584 j m n + 3483648 j m n - 19906560 j m n 9 5 5 4 6 3 7 + 1990656 j n - 1492992 m n + 7464960 m n - 8957952 m n 2 8 9 9 8 8 7 2 - 5971968 m n + 11943936 m n - 31 j - 636 j m + 606 j n - 4992 j m 7 7 2 6 3 6 2 + 14496 j m n - 1248 j n - 18688 j m + 117312 j m n 6 2 6 3 5 4 5 3 - 114528 j m n - 69088 j n - 33024 j m + 413952 j m n 5 2 2 5 3 5 4 4 5 - 1136448 j m n + 127104 j m n + 894048 j n - 21504 j m 4 4 4 3 2 4 2 3 4 4 + 632064 j m n - 3892224 j m n + 5725824 j m n + 3991296 j m n 4 5 3 5 3 4 2 3 3 3 - 5300544 j n + 322560 j m n - 5022720 j m n + 20160000 j m n 3 2 4 3 5 3 6 2 5 2 - 15022080 j m n - 31233024 j m n + 17132544 j n - 1935360 j m n 2 4 3 2 3 4 2 2 5 + 21219840 j m n - 62173440 j m n + 11688192 j m n 2 6 2 7 5 3 4 4 + 108497664 j m n - 28829952 j n + 5806080 j m n - 50284800 j m n 3 5 2 6 7 8 + 105090048 j m n + 32306688 j m n - 185213952 j m n + 19243008 j n 5 4 4 5 3 6 2 7 - 8709120 m n + 51632640 m n - 70668288 m n - 57977856 m n 8 9 8 7 7 6 2 + 124416000 m n + 995328 n + 302 j + 7640 j m - 304 j n + 63616 j m 6 6 2 5 3 5 2 - 115024 j m n - 129752 j n + 228608 j m - 1221312 j m n 5 2 5 3 4 4 4 3 + 52800 j m n + 2095424 j n + 353024 j m - 4294144 j m n 4 2 2 4 3 4 4 3 5 + 9070464 j m n + 10057856 j m n - 15181216 j n + 179200 j m 3 4 3 3 2 3 2 3 - 5608960 j m n + 33326080 j m n - 30412160 j m n 3 4 3 5 2 5 2 4 2 - 92325760 j m n + 57977728 j n - 2150400 j m n + 35535360 j m n 2 3 3 2 2 4 2 5 - 136888320 j m n + 21410880 j m n + 377765760 j m n 2 6 5 2 4 3 - 111498816 j n + 9676800 j m n - 112250880 j m n 3 4 2 5 6 7 + 288645120 j m n + 131272704 j m n - 745445376 j m n + 80289792 j n 5 3 4 4 3 5 2 6 - 19353600 m n + 144184320 m n - 231026688 m n - 246786048 m n 7 8 7 6 6 + 568442880 m n + 10838016 n + 498 j - 34616 j m - 109964 j n 5 2 5 5 2 4 3 - 416064 j m - 137472 j m n + 2451336 j n - 1523200 j m 4 2 4 2 4 3 3 4 + 6000000 j m n + 12811200 j m n - 22971280 j n - 2016000 j m 3 3 3 2 2 3 3 + 23600640 j m n - 27944640 j m n - 145002240 j m n 3 4 2 5 2 4 2 3 2 + 107508000 j n - 752640 j m + 25509120 j m n - 145178880 j m n 2 2 3 2 4 2 5 5 + 7403520 j m n + 723716160 j m n - 241888128 j n + 6773760 j m n 4 2 3 3 2 4 - 120718080 j m n + 407877120 j m n + 297414720 j m n 5 6 5 2 4 3 - 1693471104 j m n + 188547264 j n - 20321280 m n + 207567360 m n 3 4 2 5 6 7 - 404248320 m n - 599446656 m n + 1493655552 m n + 51767424 n 6 5 5 4 2 4 - 34935 j - 100812 j m + 1424508 j n + 1372000 j m + 8140040 j m n 4 2 3 3 3 2 3 2 - 19313312 j n + 6016640 j m - 11001984 j m n - 127103616 j m n 3 3 2 4 2 3 2 2 2 + 117692192 j n + 6594560 j m - 73862656 j m n - 19638528 j m n 2 3 2 4 5 4 + 822080768 j m n - 320989936 j n + 1662976 j m - 62261248 j m n 3 2 2 3 4 + 311594752 j m n + 398814976 j m n - 2371328896 j m n 5 5 4 2 3 3 + 271485568 j n - 9977856 m n + 161839104 m n - 407404032 m n 2 4 5 6 5 - 903775488 m n + 2484084480 m n + 142528512 n + 326769 j 4 4 3 2 3 + 2037700 j m - 8517482 j n - 1232448 j m - 58646112 j m n 3 2 2 3 2 2 2 2 + 75779064 j n - 14355712 j m - 21693504 j m n + 551904864 j m n 2 3 4 3 2 2 - 265558960 j n - 12235776 j m + 121524480 j m n + 312177600 j m n 3 4 5 4 - 2091247872 j m n + 243207504 j n - 1806336 m + 64382976 m n 3 2 2 3 4 5 - 235807488 m n - 860162688 m n + 2706790464 m n + 249509088 n 4 3 3 2 2 - 1530376 j - 11059040 j m + 26430256 j n - 6435456 j m 2 2 2 3 2 + 201932544 j m n - 132823152 j n + 18821632 j m + 130679808 j m n 2 3 4 3 - 1131148224 j m n + 129333184 j n + 10100736 m - 72526848 m n 2 2 3 4 3 - 500829696 m n + 1928409984 m n + 288102720 n + 3824572 j 2 2 2 + 30912496 j m - 36341768 j n + 22366464 j m - 341995584 j m n 2 3 2 2 + 34399440 j n - 9169920 m - 161708544 m n + 864278208 m n 3 2 2 + 219293856 n - 4099376 j - 44097728 j m + 1482112 j n - 21934080 m 2 + 220718208 m n + 105912768 n - 816768 j + 24454656 m + 29355264 n 4 + 3538944) F(n + 1, n0) + 3 (j - 3 n - 6) (n + 2) (3 n + 4) (3 n + 5) (j 3 3 2 2 2 2 2 2 + 8 j m - 16 j n + 16 j m - 112 j m n + 88 j n - 144 j m n 2 3 2 2 3 3 2 + 576 j m n - 144 j n + 432 m n - 864 m n - 10 j - 72 j m 2 2 2 2 2 3 + 108 j n - 96 j m + 744 j m n - 240 j n + 576 m n - 1656 m n - 72 n 2 2 2 + 31 j + 232 j m - 116 j n + 176 m - 1040 m n - 172 n - 14 j - 216 m - 132 n - 32) (j - 3 n - 4) (j - 3 n - 5) F(n + 2, n0) = G(n, n0 + 1) - G(n, n0), (Check!) and the theorem follows upon summing with respect to, n0, . QED. For the variable n, and F2, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(n, k), be given by (n + m) k (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k) and let, SUM(n), be the sum of, F(n, k), with respect to, k SUM(n), satisfies the following linear recurrence equation -128 (2 m + 2 n + 1) (m + n + 1) (j + 2 m - 4 n - 2) (j + 2 m - 4 n - 4) 4 3 3 2 2 (j + 2 m - 4 n - 1) (j + 2 m - 4 n - 3) (j + 8 j m - 16 j n + 16 j m 2 2 2 2 2 3 2 2 - 112 j m n + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n 3 3 2 2 2 - 864 m n - 26 j - 184 j m + 284 j n - 240 j m + 1896 j m n 2 2 2 3 2 - 672 j n + 1440 m n - 4248 m n - 72 n + 227 j + 1552 j m - 1028 j n 2 2 + 1184 m - 6944 m n - 388 n - 514 j - 3776 m - 692 n - 408) SUM(n) + ( 10 9 9 8 2 8 8 2 7 3 j + 20 j m - 28 j n + 160 j m - 568 j m n + 280 j n + 640 j m 7 2 7 2 7 3 6 4 6 3 - 4416 j m n + 6528 j m n - 512 j n + 1280 j m - 16384 j m n 6 2 2 6 3 6 4 5 5 + 52032 j m n - 35200 j m n - 14176 j n + 1024 j m 5 4 5 3 2 5 2 3 5 4 - 28672 j m n + 181504 j m n - 337664 j m n + 39296 j m n 5 5 4 5 4 4 2 4 3 3 + 152704 j n - 18432 j m n + 274176 j m n - 1138176 j m n 4 2 4 4 5 4 6 3 5 2 + 1287936 j m n + 647424 j m n - 764928 j n + 138240 j m n 3 4 3 3 3 4 3 2 5 3 6 - 1451520 j m n + 4423680 j m n - 2764800 j m n - 4396032 j m n 3 7 2 5 3 2 4 4 2 3 5 + 2142720 j n - 552960 j m n + 4596480 j m n - 10920960 j m n 2 2 6 2 7 2 8 5 4 + 2059776 j m n + 13243392 j m n - 3207168 j n + 1244160 j m n 4 5 3 6 2 7 8 - 8709120 j m n + 15427584 j m n + 3483648 j m n - 19906560 j m n 9 5 5 4 6 3 7 + 1990656 j n - 1492992 m n + 7464960 m n - 8957952 m n 2 8 9 9 8 8 7 2 - 5971968 m n + 11943936 m n - 31 j - 636 j m + 606 j n - 4992 j m 7 7 2 6 3 6 2 + 14496 j m n - 1248 j n - 18688 j m + 117312 j m n 6 2 6 3 5 4 5 3 - 114528 j m n - 69088 j n - 33024 j m + 413952 j m n 5 2 2 5 3 5 4 4 5 - 1136448 j m n + 127104 j m n + 894048 j n - 21504 j m 4 4 4 3 2 4 2 3 4 4 + 632064 j m n - 3892224 j m n + 5725824 j m n + 3991296 j m n 4 5 3 5 3 4 2 3 3 3 - 5300544 j n + 322560 j m n - 5022720 j m n + 20160000 j m n 3 2 4 3 5 3 6 2 5 2 - 15022080 j m n - 31233024 j m n + 17132544 j n - 1935360 j m n 2 4 3 2 3 4 2 2 5 + 21219840 j m n - 62173440 j m n + 11688192 j m n 2 6 2 7 5 3 4 4 + 108497664 j m n - 28829952 j n + 5806080 j m n - 50284800 j m n 3 5 2 6 7 8 + 105090048 j m n + 32306688 j m n - 185213952 j m n + 19243008 j n 5 4 4 5 3 6 2 7 - 8709120 m n + 51632640 m n - 70668288 m n - 57977856 m n 8 9 8 7 7 6 2 + 124416000 m n + 995328 n + 302 j + 7640 j m - 304 j n + 63616 j m 6 6 2 5 3 5 2 - 115024 j m n - 129752 j n + 228608 j m - 1221312 j m n 5 2 5 3 4 4 4 3 + 52800 j m n + 2095424 j n + 353024 j m - 4294144 j m n 4 2 2 4 3 4 4 3 5 + 9070464 j m n + 10057856 j m n - 15181216 j n + 179200 j m 3 4 3 3 2 3 2 3 - 5608960 j m n + 33326080 j m n - 30412160 j m n 3 4 3 5 2 5 2 4 2 - 92325760 j m n + 57977728 j n - 2150400 j m n + 35535360 j m n 2 3 3 2 2 4 2 5 - 136888320 j m n + 21410880 j m n + 377765760 j m n 2 6 5 2 4 3 - 111498816 j n + 9676800 j m n - 112250880 j m n 3 4 2 5 6 7 + 288645120 j m n + 131272704 j m n - 745445376 j m n + 80289792 j n 5 3 4 4 3 5 2 6 - 19353600 m n + 144184320 m n - 231026688 m n - 246786048 m n 7 8 7 6 6 + 568442880 m n + 10838016 n + 498 j - 34616 j m - 109964 j n 5 2 5 5 2 4 3 - 416064 j m - 137472 j m n + 2451336 j n - 1523200 j m 4 2 4 2 4 3 3 4 + 6000000 j m n + 12811200 j m n - 22971280 j n - 2016000 j m 3 3 3 2 2 3 3 + 23600640 j m n - 27944640 j m n - 145002240 j m n 3 4 2 5 2 4 2 3 2 + 107508000 j n - 752640 j m + 25509120 j m n - 145178880 j m n 2 2 3 2 4 2 5 5 + 7403520 j m n + 723716160 j m n - 241888128 j n + 6773760 j m n 4 2 3 3 2 4 - 120718080 j m n + 407877120 j m n + 297414720 j m n 5 6 5 2 4 3 - 1693471104 j m n + 188547264 j n - 20321280 m n + 207567360 m n 3 4 2 5 6 7 - 404248320 m n - 599446656 m n + 1493655552 m n + 51767424 n 6 5 5 4 2 4 - 34935 j - 100812 j m + 1424508 j n + 1372000 j m + 8140040 j m n 4 2 3 3 3 2 3 2 - 19313312 j n + 6016640 j m - 11001984 j m n - 127103616 j m n 3 3 2 4 2 3 2 2 2 + 117692192 j n + 6594560 j m - 73862656 j m n - 19638528 j m n 2 3 2 4 5 4 + 822080768 j m n - 320989936 j n + 1662976 j m - 62261248 j m n 3 2 2 3 4 + 311594752 j m n + 398814976 j m n - 2371328896 j m n 5 5 4 2 3 3 + 271485568 j n - 9977856 m n + 161839104 m n - 407404032 m n 2 4 5 6 5 - 903775488 m n + 2484084480 m n + 142528512 n + 326769 j 4 4 3 2 3 + 2037700 j m - 8517482 j n - 1232448 j m - 58646112 j m n 3 2 2 3 2 2 2 2 + 75779064 j n - 14355712 j m - 21693504 j m n + 551904864 j m n 2 3 4 3 2 2 - 265558960 j n - 12235776 j m + 121524480 j m n + 312177600 j m n 3 4 5 4 - 2091247872 j m n + 243207504 j n - 1806336 m + 64382976 m n 3 2 2 3 4 5 - 235807488 m n - 860162688 m n + 2706790464 m n + 249509088 n 4 3 3 2 2 - 1530376 j - 11059040 j m + 26430256 j n - 6435456 j m 2 2 2 3 2 + 201932544 j m n - 132823152 j n + 18821632 j m + 130679808 j m n 2 3 4 3 - 1131148224 j m n + 129333184 j n + 10100736 m - 72526848 m n 2 2 3 4 3 - 500829696 m n + 1928409984 m n + 288102720 n + 3824572 j 2 2 2 + 30912496 j m - 36341768 j n + 22366464 j m - 341995584 j m n 2 3 2 2 + 34399440 j n - 9169920 m - 161708544 m n + 864278208 m n 3 2 2 + 219293856 n - 4099376 j - 44097728 j m + 1482112 j n - 21934080 m 2 + 220718208 m n + 105912768 n - 816768 j + 24454656 m + 29355264 n + 3538944) SUM(n + 1) + 3 (3 n + 5) (3 n + 4) (n + 2) (j - 3 n - 4) 4 3 3 2 2 2 (j - 3 n - 5) (j - 3 n - 6) (j + 8 j m - 16 j n + 16 j m - 112 j m n 2 2 2 2 3 2 2 3 + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n - 864 m n 3 2 2 2 2 2 - 10 j - 72 j m + 108 j n - 96 j m + 744 j m n - 240 j n + 576 m n 2 3 2 2 - 1656 m n - 72 n + 31 j + 232 j m - 116 j n + 176 m - 1040 m n 2 - 172 n - 14 j - 216 m - 132 n - 32) SUM(n + 2) =0. PROOF: We cleverly construct, G(n, k), := 8 7 6 5 4 - 4 (-456966144 + 169728 k - 1070592 k - 509184 k + 5548800 k - 894720 k 3 2 9 11 10 12 - 8838144 k + 36194304 k + 65280 k + 575488 m - 13056 k - 37888 m 8 7 6 5 + (324753408 - 102784 k - 1322624 k + 2903232 k + 2766784 k 4 3 2 9 11 - 10980928 k + 2421120 k - 6765248 k + 127936 k - 274432 m 10 12 13 11 - 16448 k + 7680 m - 19501056 n + (-117504 m 10 2 9 + (-48384 k + 4119552) m + (488448 k + 1399680 k - 43105408) m 3 2 8 + (193536 k - 14826240 k + 1325376 k + 223945056) m 4 3 2 7 + (-778752 k - 4582656 k + 143887872 k - 129392064 k - 832623072) m + 5 4 3 2 (-290304 k + 19782912 k + 3982208 k - 699948352 k + 797180512 k 6 6 5 4 3 + 2993917392) m + (580608 k + 5349888 k - 167820288 k + 199460544 k 2 5 7 + 2217370464 k - 2702215776 k - 13974092736) m + (193536 k 6 5 4 3 2 - 11586816 k - 11789568 k + 710350464 k - 1189908384 k - 5065097808 k 4 8 7 6 + 7658169792 k + 49089056160) m + (-191232 k - 2550528 k + 76398592 k 5 4 3 2 - 66740544 k - 1776036000 k + 3401358080 k + 5918687904 k 3 9 8 7 - 14777661088 k - 30457628992) m + (-48384 k + 2532096 k + 6331008 k 6 5 4 3 - 235749312 k + 308882208 k + 2889929520 k - 5120893632 k 2 2 10 + 2456159088 k + 10016947344 k - 211506561888) m + (18432 k 9 8 7 6 5 + 383616 k - 9360768 k - 3327936 k + 350622240 k - 438979488 k 4 3 2 - 2681434080 k + 3583936800 k - 8948167152 k + 4003096656 k 10 9 8 7 + 309893313936) m - 21504 k + 150976 k + 1402144 k - 10400480 k 6 5 4 3 2 - 2014640 k + 106652032 k - 125987920 k - 193211360 k + 371601248 k 2 9 8 - 862797248 k + 5135976992) n + (-3997440 m + (-1099008 k + 81609984) m 2 7 + (12704256 k - 6753024 k - 773886912) m 3 2 6 + (3566592 k - 231826176 k + 247784448 k + 4451268960) m + 4 3 2 (-14070528 k - 10589184 k + 1824227328 k - 2174539392 k - 23217475776) 5 5 4 3 2 m + (-3836160 k + 214343040 k - 260113984 k - 7941632224 k 4 6 5 + 12178510288 k + 104476657680) m + (6018048 k + 24717312 k 4 3 2 - 1276677952 k + 2144655232 k + 17492453760 k - 37906137008 k 3 7 6 5 - 107190546616) m + (1368576 k - 64717056 k - 38625408 k 4 3 2 + 3877313952 k - 5731244064 k - 8839781520 k + 45613795488 k 2 8 7 6 5 - 733026767352) m + (-654336 k - 7375104 k + 211454720 k - 10325376 k 4 3 2 - 5559003648 k + 5557693360 k - 17647478232 k - 436400360 k 8 7 6 5 + 1496693988448) m + 590208 k - 3255616 k - 32881824 k + 168387344 k 4 3 2 + 146860992 k - 1124820272 k + 2752256672 k - 4432519984 k 4 10 9 8 7 - 29535282712) n + (49664 k + 174592 k - 3727488 k + 5253760 k 6 5 4 3 2 + 33771840 k - 63744896 k - 89098752 k + 167367040 k - 355012544 k 7 + 209717888 k + 6743581824) m + (-38527488 m 6 + (-1036800 k + 671733504) m 2 5 + (83856384 k - 115036416 k - 5790808704) m 3 2 4 + (8474112 k - 1205581824 k + 2370866880 k + 39274441056) m + 4 3 2 (-53303040 k + 36205056 k + 6060298368 k - 15222303264 k - 70228169424) 3 5 4 3 2 m + (-9476352 k + 526677888 k - 466996480 k - 9706900864 k 2 6 5 + 33199156096 k - 559143422568) m + (7686144 k + 52721280 k 4 3 2 - 1572525440 k + 588452864 k - 3759825408 k - 11755857280 k 6 5 4 + 1683669096280) m - 6355200 k + 25757376 k + 253042496 k 3 2 6 - 780086480 k + 2406869208 k - 3834038648 k - 114008792224) n + ( 6 5 79612416 m + (-2571264 k - 1275872256) m 2 4 + (-142110720 k + 413344512 k + 12300069504) m 3 2 3 + (-9497088 k + 1499132160 k - 4709334336 k - 31072788576) m + 4 3 2 (59164416 k + 21921408 k - 4153302336 k + 14900045472 k - 289471264464) 2 5 4 3 2 m + (8667648 k - 359922432 k - 163411136 k - 15681440 k - 8689635856 k 6 5 4 3 + 1096194495672) m - 778752 k + 3058560 k + 114455680 k - 312751168 k 2 7 5 + 1133644128 k - 1936959904 k - 110331411560) n + (-121036032 m 4 + (26936064 k + 2205802368) m 2 3 + (156280320 k - 805545216 k - 8779243584) m 3 2 2 + (15040512 k - 957899520 k + 4170541824 k - 103798455552) m + 4 3 2 (-34993152 k - 108615168 k + 456234240 k - 3864493440 k + 513637134240) 4 3 2 m + 27299712 k - 70172608 k + 318242528 k - 632092928 k - 74368085544) 8 3 2 n + (-102168576 m + (43877376 k - 3462309504) m 2 2 + (16109568 k - 149617152 k + 37142123776) m + 284160 k - 12033920 k 10 8 7 - 11637143136) n + (13208832 m + (1078272 k - 272885760) m 2 6 + (-36149760 k + 34767360 k + 2359610304) m 3 2 5 + (-6041088 k + 616128768 k - 759506112 k - 14782823712) m + 4 3 2 4 (32097024 k - 6417792 k - 4237011264 k + 7052469408 k + 79169179920) m 5 4 3 2 + (6428160 k - 409721088 k + 591346432 k + 13389565120 k 3 6 5 - 30125267632 k - 105752900952) m + (-9248256 k - 44198784 k 4 3 2 + 1950348288 k - 2497126656 k - 12880055808 k + 48099011280 k 2 7 6 5 - 762457014744) m + (-1465344 k + 63152640 k + 100279872 k 4 3 2 - 3804697824 k + 2914626288 k - 11408640888 k - 8378340168 k 8 7 6 5 + 1873257640968) m + 92160 k - 490752 k - 20629760 k + 90084960 k 4 3 2 + 301030224 k - 1187673808 k + 3232539192 k - 5051951992 k 5 4 3 - 77872080952) n + (172419840 m + (-57438720 k - 1431098880) m 2 2 + (-92689920 k + 656505216 k - 24578333824) m 3 2 + (-16312320 k + 149530368 k - 1029524288 k + 169072894176) m 4 3 2 9 + 2700288 k - 6759936 k + 43555200 k - 123460096 k - 35317109536) n + 10 9 (946944 m + (414720 k - 21649920) m 2 8 + (-3359232 k - 2930688 k + 195161280) m 3 2 7 + (-1340928 k + 70407936 k - 57353280 k - 1110950496) m 4 3 2 6 + (4400640 k + 11456640 k - 579044928 k + 641297760 k + 4836234864) m 5 4 3 2 + (1534464 k - 79908096 k + 58393408 k + 2760551200 k - 3279402688 k 5 6 5 4 - 22970533968) m + (-2515968 k - 14090112 k + 559527872 k 3 2 4 - 852218912 k - 8481594000 k + 12613182640 k + 90210971840) m + ( 7 6 5 4 3 -705024 k + 35192064 k + 10818624 k - 2051517728 k + 3703502912 k 2 3 8 + 13587740736 k - 30293210560 k - 71952076592) m + (532224 k 7 6 5 4 3 + 5533056 k - 177039296 k + 123818400 k + 4438038288 k - 7194611680 k 2 2 9 - 1216389024 k + 27636841760 k - 486197479384) m + (96768 k 8 7 6 5 4 - 4041984 k - 11858752 k + 368286816 k - 298207744 k - 4999096560 k 3 2 + 5862140320 k - 16231942768 k + 4641686864 k + 836411763128) m 10 9 8 7 6 - 4608 k + 31104 k + 1395072 k - 8386784 k - 24043376 k 5 4 3 2 + 179504752 k - 65282272 k - 640194448 k + 1411734352 k - 2516949248 k 3 12 2 - 571509664) n + (292515840 m - 330846720) n + (-220022784 m 2 11 + (-8902656 k + 4892963328) m - 423936 k - 317952 k - 2539806208) n + ( 10 9 8 7 6 5 -7680 k - 94208 k + 2119744 k - 2211584 k - 37740032 k + 75319936 k 4 3 2 + 122134592 k - 298457216 k + 312225920 k + 120119680 k - 6176632000) 2 8 7 6 5 4 m + (-473088 k - 765440 k + 25792768 k - 44016512 k - 127857856 k 3 2 3 6 + 300467456 k + 62961664 k - 443502464 k - 743551552) m + (1395712 k 5 4 3 2 + 1248768 k - 52182528 k + 78877568 k + 121590720 k - 188569984 k 5 8 7 6 5 - 699380224) m + (38400 k + 376832 k - 9011328 k + 9698432 k 4 3 2 + 102241536 k - 196267904 k - 155562688 k + 321938624 k + 2053214144) 4 6 5 4 3 2 m + (-76800 k - 565248 k + 14480000 k - 13273856 k - 88720896 k 6 + 111670144 k + 118851008) m 4 3 2 8 + (76800 k + 376832 k - 10322752 k + 5659072 k + 24322176) m 4 3 2 7 + (-1845248 k - 899584 k + 41896704 k - 40114816 k - 33488960) m 2 10 + (-38400 k - 94208 k + 2750784) m 2 9 12 11 + (1147392 k + 241664 k - 11779456) m + (4608 m - 361728 m 2 10 + (-23040 k - 138240 k + 5879680) m 2 9 + (1507584 k + 1316608 k - 37294848) m 4 3 2 8 + (46080 k + 552960 k - 21555840 k + 8631360 k + 121579296) m 4 3 2 7 + (-2413056 k - 4424192 k + 128220928 k - 120051136 k - 294560928) m 6 5 4 3 2 + (-46080 k - 829440 k + 29422336 k - 16572928 k - 405630336 k 6 6 5 4 + 479668160 k + 960236512) m + (1810944 k + 5372928 k - 154196736 k 3 2 5 8 + 215258688 k + 870237984 k - 1138704384 k - 4773615504) m + (23040 k 7 6 5 4 3 + 552960 k - 17728768 k + 7493760 k + 435697344 k - 779225792 k 2 4 8 - 1507109280 k + 2477324800 k + 15246531088) m + (-604416 k 7 6 5 4 3 - 2739712 k + 72910080 k - 102925120 k - 772695648 k + 1597075840 k 2 3 10 9 + 1214486240 k - 3969988000 k - 7292496496) m + (-4608 k - 138240 k 8 7 6 5 4 + 4015488 k + 205824 k - 152067456 k + 256369728 k + 968606112 k 3 2 2 - 1932678528 k + 1606150560 k + 1872084000 k - 54190012176) m + ( 10 9 8 7 6 60672 k + 474368 k - 9639424 k + 7717568 k + 172119008 k 5 4 3 2 - 270887296 k - 773454416 k + 1191780128 k - 2731405136 k 10 9 8 + 1471173536 k + 68242017184) m - 32896 k + 241984 k + 421152 k 7 6 5 4 3 - 6116672 k + 7085088 k + 30946432 k - 62674224 k - 18499232 k 2 8 + 21443824 k - 153003616 k + 2231997312) n - 9158784 k) j + (-8792064 m 7 2 6 + 188490240 m + (24634368 k - 24634368 k - 1614914304) m 2 5 + (-425841408 k + 425841408 k + 11101791936) m + 4 3 2 (-22546944 k + 45093888 k + 2919275520 k - 2941822464 k - 62494366848) 4 m + 4 3 2 (280396800 k - 560793600 k - 8727546432 k + 9007943232 k + 47350229232) 3 6 5 4 3 m + (6704640 k - 20113920 k - 1274676352 k + 2582875904 k 2 2 6 + 4323219712 k - 5618009984 k + 771767967936) m + (-40856832 k 5 4 3 2 + 122570496 k + 2150974336 k - 4506232832 k + 17496325808 k 6 5 - 15222780976 k - 1568819539776) m - 3618944 k + 10856832 k 4 3 2 + 209429376 k - 436953472 k + 1824877760 k - 1604591552 k 6 5 4 - 169994927024) n + (-5132 n + (5244 k - 196 m - 39802) n 2 2 3 + (1720 m + (-5424 k + 1184) m - 2676 k + 30942 k - 122012) n + ( 3 2 2 3 -2000 m + (1584 k + 6474) m + (3684 k - 25974 k + 8244) m + 524 k 2 2 4 3 - 10956 k + 67108 k - 184688) n + (1055 m + (348 k - 5200) m 2 2 3 2 + (-1362 k + 6012 k + 7530) m + (-668 k + 10644 k - 40336 k + 13739) m 4 3 2 5 - 13 k + 1232 k - 14474 k + 63289 k - 137960) n - 161 m 4 2 3 + (-159 k + 1448) m + (162 k + 126 k - 3143) m 3 2 2 + (158 k - 2070 k + 5299 k + 2632) m 4 3 2 5 4 3 + (-k - 766 k + 7351 k - 20264 k + 7048) m + k - 18 k + 701 k 2 9 7 6 - 6144 k + 21840 k - 40656) j + (22975488 m - 386836992 m 2 5 + (-50761728 k + 50761728 k + 3592700928) m 2 4 + (718288128 k - 718288128 k - 26236994880) m + 4 3 2 3 (32804352 k - 65608704 k - 3435868416 k + 3468672768 k + 28418408256) m + ( 4 3 2 -310961664 k + 621923328 k + 3866546112 k - 4177507776 k + 489469120320) 2 6 5 4 3 m + (-4672512 k + 14017536 k + 821300608 k - 1665963776 k 2 6 5 + 7469608448 k - 6634290304 k - 1215594371184) m - 389376 k + 1168128 k 4 3 2 + 76048960 k - 154044800 k + 719198096 k - 641981008 k - 125566050384) 7 10 8 7 6 5 n + (434658720 - 3104 m + 10280 k - 243628 k + 459470 k + 2443334 k 4 3 2 9 10 10 - 7477014 k + 199594 k - 7178556 k + 2752 k + 32 k + 29930176 n 4 3 2 6 + (-5696 k - 294784 k + 1922064 k + 264412 k - 21329198) m 9 2 8 + (-5248 k + 80736) m + (8096 k + 115520 k - 908904) m 3 2 7 9 + (16640 k - 161408 k - 683376 k + 5822988) m + (58528 m 8 2 7 + (83520 k - 1432416) m + (-112448 k - 1117856 k + 14886456) m 3 2 6 + (-209344 k + 2857792 k + 2719296 k - 77132736) m 4 3 2 5 + (38400 k + 2798688 k - 29584584 k + 28917984 k + 226625442) m + ( 5 4 3 2 169920 k - 1252224 k - 12756128 k + 148376248 k - 262608972 k 4 6 5 4 3 - 669044561) m + (26432 k - 1949664 k + 13227048 k + 14920912 k 2 3 7 6 - 413178172 k + 981124856 k + 1464965108) m + (-45888 k - 191040 k 5 4 3 2 + 9844416 k - 49108416 k + 2657496 k + 563428686 k - 1779295194 k 2 8 7 6 5 + 1809475233) m + (-10912 k + 268832 k + 629096 k - 19737520 k 4 3 2 + 57606122 k + 29782944 k - 137170032 k + 1239429650 k - 9011070310) m 9 8 7 6 5 4 + 1792 k + 17888 k - 494432 k + 892616 k + 10771428 k - 33605109 k 3 2 8 - 10498490 k - 33214859 k - 154377722 k + 3605688512) n + (-554112 m 7 2 6 + (-449280 k + 12314640) m + (1052864 k + 3703328 k - 100507032) m 3 2 5 + (1062528 k - 22751376 k + 17868528 k + 396578772) m 4 3 2 4 + (-410304 k - 11142240 k + 180050256 k - 362778552 k - 1345282590) m 5 4 3 2 + (-705024 k + 9140976 k + 32735888 k - 712751572 k + 1911828054 k 3 6 5 4 3 + 3538658036) m + (-95424 k + 7319520 k - 51416712 k - 47544708 k 2 2 7 6 + 1310584494 k - 4400415600 k + 4109745936) m + (91776 k + 681360 k 5 4 3 2 - 23073408 k + 79270896 k + 121985826 k - 519445968 k + 3802527366 k 8 7 6 5 - 27564237582) m + 6976 k - 339872 k + 513232 k + 18474604 k 4 3 2 2 - 61324088 k - 55952508 k - 47862330 k - 634817546 k + 13145080696) n 7 6 + (3312192 m + (1340160 k - 56332704) m 2 5 + (-5754752 k + 640288 k + 336860184) m 3 2 4 + (-3164480 k + 94919024 k - 235752096 k - 1423295088) m 4 3 2 3 + (2078912 k + 24620224 k - 593564856 k + 1920936104 k + 4656473426) m 5 4 3 2 + (1806336 k - 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476223936 k + 1770344976 k 6 5 4 - 15914475420 k + 257527256058) m - 414592 k - 13336992 k + 127158656 k 3 2 5 + 183223696 k + 63822500 k + 2841340400 k - 125617215236) n + ( 7 6 -12729216 m + (-2690688 k + 212712192) m 2 5 + (24808128 k - 30151200 k - 1419562512) m 3 2 4 + (9179200 k - 376520480 k + 974841808 k + 7680833976) m + 4 3 2 (-12132416 k - 52081024 k + 2190787104 k - 7472878040 k - 22765335220) 3 5 4 3 2 m + (-6810048 k + 124890432 k + 133494336 k - 5085413928 k 2 6 5 + 21520141080 k - 53128901646) m + (325888 k + 47474208 k 4 3 2 - 321265472 k - 706206808 k + 1994349712 k - 19220913200 k 7 6 5 4 + 286073336258) m + 248320 k - 3396000 k - 35234912 k + 278241104 k 3 2 4 + 112805644 k + 398255010 k + 2357241674 k - 113261070422) n 10 2 + (3980288 k + 203689472 m - 734237664) n + (-54985280 m 2 9 + (-43099264 k + 2870858880) m - 1895040 k + 54167584 k - 4994861952) n 3 2 + (-102242112 m + (97780032 k - 762568128) m 2 3 2 + (22776576 k - 529459680 k + 18034989408) m + 4571392 k - 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2010272 k + 23069378 k - 49341914 k - 52906613) m 5 4 3 2 4 + (-8160 k - 800 k + 829740 k - 7298084 k + 11846025 k + 23970428) m 4 3 2 5 + (3616 k - 153072 k + 1273356 k - 546976 k - 11322185) m + ( 2 2 4421712 m + (-13844992 k + 239150784) m - 724896 k + 33002288 k 7 8 8 - 836423464) n + (3084256 k + 20558688 m - 150476256) n + (53328 m 7 2 6 + (56480 k - 1252608) m + (-77184 k - 650248 k + 10669040) m 3 2 5 + (-112992 k + 1977504 k + 319296 k - 39949884) m 4 3 2 4 + (-1120 k + 1355640 k - 17508120 k + 31359640 k + 99771245) m + 5 4 3 2 (60384 k - 473984 k - 5841648 k + 77021212 k - 187509018 k - 255734314) 3 6 5 4 3 2 m + (26880 k - 642840 k + 3079056 k + 12469236 k - 156299808 k 2 7 6 5 + 447557376 k - 109958436) m + (-3872 k - 160032 k + 2028464 k 4 3 2 - 4886256 k - 14838822 k + 91567906 k - 446246340 k + 1770465813) m 8 7 6 5 4 3 - 1904 k + 22696 k + 199064 k - 2249916 k + 3492179 k + 10010088 k 2 8 - 2527208 k + 107475723 k - 1148920056) n + (-5168 k + 67944) m 2 7 + (2688 k + 72656 k - 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1512176 k - 31634196 k + 258152050 k - 1486684440 k 6 5 4 3 + 6696357674) m + 45600 k - 1260792 k + 1842260 k + 32199586 k 2 3 5 - 40374466 k + 581601880 k - 6450853508) n + 14165716 k) j + ( 4 3 2 2 -76308480 m + 432472320 m + (34338816 k - 34338816 k + 15454454016) m 2 2 + (-626688 k + 626688 k - 86790664000) m - 381952 k + 381952 k 10 11 10 - 7937515712) n + (179712 m - 5225472 m 2 9 + (-746496 k + 746496 k + 48184448) m 2 8 + (18959616 k - 18959616 k - 258285696) m 4 3 2 7 + (1188864 k - 2377728 k - 164529408 k + 165718272 k + 1144774272) m + 4 3 2 6 (-25523712 k + 51047424 k + 801741312 k - 827265024 k - 4660211520) m 6 5 4 3 2 + (-884736 k + 2654208 k + 194446336 k - 393316352 k - 2830715392 k 5 6 5 + 3027815936 k + 23449539264) m + (15068160 k - 45204480 k 4 3 2 - 795770368 k + 1666881536 k + 7036872448 k - 7877847296 k 4 8 7 6 5 - 79516881456) m + (290304 k - 1161216 k - 88042240 k + 268190976 k 4 3 2 + 1978725504 k - 4405790720 k - 7419034880 k + 9666822272 k 3 8 7 6 + 18503383520) m + (-3276288 k + 13105152 k + 251357696 k 5 4 3 2 - 799941120 k - 3029319744 k + 7407164032 k - 8934775040 k 2 10 9 8 + 5095685312 k + 417387837200) m + (-27648 k + 138240 k + 9940864 k 7 6 5 4 - 40592896 k - 312333568 k + 1079656448 k + 2125201984 k 3 2 - 6097466240 k + 18795974880 k - 15560492064 k - 509493035968) m 10 9 8 7 6 - 2304 k + 11520 k + 957056 k - 3897344 k - 33106304 k 5 4 3 2 + 113008000 k + 236737328 k - 666391264 k + 2454697872 k 3 7 - 2102014560 k - 65305554752) n + (-570100 n 6 + (314340 k + 309188 m - 5781454) n + 2 2 5 (275208 m + (-681840 k + 3030648) m - 113324 k + 2586666 k - 24866804) n 3 2 + (-257936 m + (485712 k + 1748376) m 2 3 2 + (422780 k - 4882062 k + 12091382) m + 38356 k - 731082 k + 8714486 k 4 4 3 - 58766422) n + (216136 m + (-116448 k - 1342928) m 2 2 + (-388748 k + 2947062 k + 4274998) m 3 2 4 3 + (-94040 k + 2230080 k - 13692958 k + 25177406) m - 7876 k + 190634 k 2 3 5 - 1825060 k + 15348652 k - 82349348) n + (-106680 m 4 2 3 + (-30168 k + 816990) m + (145228 k - 645774 k - 2548946) m 3 2 2 + (70764 k - 1574226 k + 6514446 k + 4995390) m 4 3 2 5 + (2748 k - 332490 k + 4287258 k - 18771876 k + 28890552) m - 484 k 4 3 2 2 - 26612 k + 343498 k - 2189938 k + 14863644 k - 68363494) n + ( 6 5 2 4 21860 m + (15480 k - 269136) m + (-24844 k - 29460 k + 993464) m 3 2 3 + (-18656 k + 394620 k - 1082962 k - 2093398) m 4 3 2 2 + (-36 k + 163572 k - 2060502 k + 6214128 k + 2758719) m 5 4 3 2 + (616 k + 6020 k - 381502 k + 3549586 k - 12551478 k + 17334067) m 6 5 4 3 2 + 460 k - 1936 k - 29650 k + 266932 k - 1253213 k + 7478907 k 7 6 - 31098520) n - 1468 m + (-1380 k + 27392) m 2 5 + (1492 k + 15456 k - 164486) m 3 2 4 + (1420 k - 33024 k + 12614 k + 387152) m 4 3 2 3 + (140 k - 21568 k + 259820 k - 564836 k - 630491) m 5 4 3 2 2 + (20 k - 128 k + 92020 k - 870796 k + 2156097 k + 564113) m + 6 5 4 3 2 (-164 k + 992 k + 3290 k - 142172 k + 1064299 k - 3266196 k + 4250474) 7 6 5 4 3 2 m - 60 k + 640 k - 1530 k - 11012 k + 75759 k - 270629 k + 1521522 k 7 9 8 - 5972568) j + (2944512 m - 62892288 m 2 7 + (-9870336 k + 9870336 k + 637718400) m 2 6 + (181225728 k - 181225728 k - 3755740992) m + 4 3 2 5 (11460096 k - 22920192 k - 1477285632 k + 1488745728 k + 21714458112) m + ( 4 3 2 -169880832 k + 339761664 k + 6528023808 k - 6697904640 k - 100014744336) 4 6 5 4 3 m + (-5087232 k + 15261696 k + 991626112 k - 2008688384 k 2 3 6 - 13483674112 k + 14490561920 k + 53240636448) m + (51501312 k 5 4 3 2 - 154503936 k - 2865067584 k + 5987641728 k + 15566784 k 2 8 7 6 - 3035138304 k + 882541675440) m + (552960 k - 2211840 k - 148056576 k 5 4 3 2 + 451911168 k + 3453517440 k - 7662800640 k + 26825015136 k 8 7 6 - 22917927648 k - 1495113047184) m + 46080 k - 184320 k - 13925248 k 5 4 3 2 + 42420864 k + 352798224 k - 776512928 k + 3019790752 k - 2624433424 k 5 - 170626235920) n - 30652416 k) (m + n + 1) (j + 2 m - 4 n - 1) (j + 2 m - 4 n - 4) (j + 2 m - 4 n - 3) (m + n + 1/2) (j + 2 m - 4 n - 2) (n + m) k (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k)/((-2 n + m + k - 1) (-n - m - 1 + k) (-2 n - 3 + m + j - k) (-2 n - 2 + m + j - k) (-2 n + m + j - k - 1) (-2 n - 4 + m + k) (-2 n + m + k - 3) (-2 n + m + k - 2) (-n - m - 2 + k) (n + m + 1 + k)) with the motive that -128 (2 m + 2 n + 1) (m + n + 1) (j + 2 m - 4 n - 2) (j + 2 m - 4 n - 4) 4 3 3 2 2 (j + 2 m - 4 n - 1) (j + 2 m - 4 n - 3) (j + 8 j m - 16 j n + 16 j m 2 2 2 2 2 3 2 2 - 112 j m n + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n 3 3 2 2 2 - 864 m n - 26 j - 184 j m + 284 j n - 240 j m + 1896 j m n 2 2 2 3 2 - 672 j n + 1440 m n - 4248 m n - 72 n + 227 j + 1552 j m - 1028 j n 2 2 + 1184 m - 6944 m n - 388 n - 514 j - 3776 m - 692 n - 408) F(n, k) + ( 10 9 9 8 2 8 8 2 7 3 j + 20 j m - 28 j n + 160 j m - 568 j m n + 280 j n + 640 j m 7 2 7 2 7 3 6 4 6 3 - 4416 j m n + 6528 j m n - 512 j n + 1280 j m - 16384 j m n 6 2 2 6 3 6 4 5 5 + 52032 j m n - 35200 j m n - 14176 j n + 1024 j m 5 4 5 3 2 5 2 3 5 4 - 28672 j m n + 181504 j m n - 337664 j m n + 39296 j m n 5 5 4 5 4 4 2 4 3 3 + 152704 j n - 18432 j m n + 274176 j m n - 1138176 j m n 4 2 4 4 5 4 6 3 5 2 + 1287936 j m n + 647424 j m n - 764928 j n + 138240 j m n 3 4 3 3 3 4 3 2 5 3 6 - 1451520 j m n + 4423680 j m n - 2764800 j m n - 4396032 j m n 3 7 2 5 3 2 4 4 2 3 5 + 2142720 j n - 552960 j m n + 4596480 j m n - 10920960 j m n 2 2 6 2 7 2 8 5 4 + 2059776 j m n + 13243392 j m n - 3207168 j n + 1244160 j m n 4 5 3 6 2 7 8 - 8709120 j m n + 15427584 j m n + 3483648 j m n - 19906560 j m n 9 5 5 4 6 3 7 + 1990656 j n - 1492992 m n + 7464960 m n - 8957952 m n 2 8 9 9 8 8 7 2 - 5971968 m n + 11943936 m n - 31 j - 636 j m + 606 j n - 4992 j m 7 7 2 6 3 6 2 + 14496 j m n - 1248 j n - 18688 j m + 117312 j m n 6 2 6 3 5 4 5 3 - 114528 j m n - 69088 j n - 33024 j m + 413952 j m n 5 2 2 5 3 5 4 4 5 - 1136448 j m n + 127104 j m n + 894048 j n - 21504 j m 4 4 4 3 2 4 2 3 4 4 + 632064 j m n - 3892224 j m n + 5725824 j m n + 3991296 j m n 4 5 3 5 3 4 2 3 3 3 - 5300544 j n + 322560 j m n - 5022720 j m n + 20160000 j m n 3 2 4 3 5 3 6 2 5 2 - 15022080 j m n - 31233024 j m n + 17132544 j n - 1935360 j m n 2 4 3 2 3 4 2 2 5 + 21219840 j m n - 62173440 j m n + 11688192 j m n 2 6 2 7 5 3 4 4 + 108497664 j m n - 28829952 j n + 5806080 j m n - 50284800 j m n 3 5 2 6 7 8 + 105090048 j m n + 32306688 j m n - 185213952 j m n + 19243008 j n 5 4 4 5 3 6 2 7 - 8709120 m n + 51632640 m n - 70668288 m n - 57977856 m n 8 9 8 7 7 6 2 + 124416000 m n + 995328 n + 302 j + 7640 j m - 304 j n + 63616 j m 6 6 2 5 3 5 2 - 115024 j m n - 129752 j n + 228608 j m - 1221312 j m n 5 2 5 3 4 4 4 3 + 52800 j m n + 2095424 j n + 353024 j m - 4294144 j m n 4 2 2 4 3 4 4 3 5 + 9070464 j m n + 10057856 j m n - 15181216 j n + 179200 j m 3 4 3 3 2 3 2 3 - 5608960 j m n + 33326080 j m n - 30412160 j m n 3 4 3 5 2 5 2 4 2 - 92325760 j m n + 57977728 j n - 2150400 j m n + 35535360 j m n 2 3 3 2 2 4 2 5 - 136888320 j m n + 21410880 j m n + 377765760 j m n 2 6 5 2 4 3 - 111498816 j n + 9676800 j m n - 112250880 j m n 3 4 2 5 6 7 + 288645120 j m n + 131272704 j m n - 745445376 j m n + 80289792 j n 5 3 4 4 3 5 2 6 - 19353600 m n + 144184320 m n - 231026688 m n - 246786048 m n 7 8 7 6 6 + 568442880 m n + 10838016 n + 498 j - 34616 j m - 109964 j n 5 2 5 5 2 4 3 - 416064 j m - 137472 j m n + 2451336 j n - 1523200 j m 4 2 4 2 4 3 3 4 + 6000000 j m n + 12811200 j m n - 22971280 j n - 2016000 j m 3 3 3 2 2 3 3 + 23600640 j m n - 27944640 j m n - 145002240 j m n 3 4 2 5 2 4 2 3 2 + 107508000 j n - 752640 j m + 25509120 j m n - 145178880 j m n 2 2 3 2 4 2 5 5 + 7403520 j m n + 723716160 j m n - 241888128 j n + 6773760 j m n 4 2 3 3 2 4 - 120718080 j m n + 407877120 j m n + 297414720 j m n 5 6 5 2 4 3 - 1693471104 j m n + 188547264 j n - 20321280 m n + 207567360 m n 3 4 2 5 6 7 - 404248320 m n - 599446656 m n + 1493655552 m n + 51767424 n 6 5 5 4 2 4 - 34935 j - 100812 j m + 1424508 j n + 1372000 j m + 8140040 j m n 4 2 3 3 3 2 3 2 - 19313312 j n + 6016640 j m - 11001984 j m n - 127103616 j m n 3 3 2 4 2 3 2 2 2 + 117692192 j n + 6594560 j m - 73862656 j m n - 19638528 j m n 2 3 2 4 5 4 + 822080768 j m n - 320989936 j n + 1662976 j m - 62261248 j m n 3 2 2 3 4 + 311594752 j m n + 398814976 j m n - 2371328896 j m n 5 5 4 2 3 3 + 271485568 j n - 9977856 m n + 161839104 m n - 407404032 m n 2 4 5 6 5 - 903775488 m n + 2484084480 m n + 142528512 n + 326769 j 4 4 3 2 3 + 2037700 j m - 8517482 j n - 1232448 j m - 58646112 j m n 3 2 2 3 2 2 2 2 + 75779064 j n - 14355712 j m - 21693504 j m n + 551904864 j m n 2 3 4 3 2 2 - 265558960 j n - 12235776 j m + 121524480 j m n + 312177600 j m n 3 4 5 4 - 2091247872 j m n + 243207504 j n - 1806336 m + 64382976 m n 3 2 2 3 4 5 - 235807488 m n - 860162688 m n + 2706790464 m n + 249509088 n 4 3 3 2 2 - 1530376 j - 11059040 j m + 26430256 j n - 6435456 j m 2 2 2 3 2 + 201932544 j m n - 132823152 j n + 18821632 j m + 130679808 j m n 2 3 4 3 - 1131148224 j m n + 129333184 j n + 10100736 m - 72526848 m n 2 2 3 4 3 - 500829696 m n + 1928409984 m n + 288102720 n + 3824572 j 2 2 2 + 30912496 j m - 36341768 j n + 22366464 j m - 341995584 j m n 2 3 2 2 + 34399440 j n - 9169920 m - 161708544 m n + 864278208 m n 3 2 2 + 219293856 n - 4099376 j - 44097728 j m + 1482112 j n - 21934080 m 2 + 220718208 m n + 105912768 n - 816768 j + 24454656 m + 29355264 n + 3538944) F(n + 1, k) + 3 (3 n + 5) (3 n + 4) (n + 2) (j - 3 n - 4) 4 3 3 2 2 2 (j - 3 n - 5) (j - 3 n - 6) (j + 8 j m - 16 j n + 16 j m - 112 j m n 2 2 2 2 3 2 2 3 + 88 j n - 144 j m n + 576 j m n - 144 j n + 432 m n - 864 m n 3 2 2 2 2 2 - 10 j - 72 j m + 108 j n - 96 j m + 744 j m n - 240 j n + 576 m n 2 3 2 2 - 1656 m n - 72 n + 31 j + 232 j m - 116 j n + 176 m - 1040 m n 2 - 172 n - 14 j - 216 m - 132 n - 32) F(n + 2, k) = G(n, k + 1) - G(n, k), (Check!) and the theorem follows upon summing with respect to, k, . QED. One can see that these are recurrences are identical -------------------------------------------- -------------------------------------------- For the variable j, and F1, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(j, n0), be given by n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n) and let, SUM(j), be the sum of, F(j, n0), with respect to, n0 SUM(j), satisfies the following linear recurrence equation (j - 3 n) SUM(j) + (-3 j - 4 m + 8 n - 2) SUM(j + 1) + (2 j + 4 m - 8 n + 2) SUM(j + 2) =0. PROOF: We cleverly construct, G(j, n0), := n0 3 n n0 (-2) binomial(2 n + 2 m, n0) binomial(6 n - j - n0, 3 n) - ----------------------------------------------------------------- -6 n + j + n0 with the motive that (j - 3 n) F(j, n0) + (-3 j - 4 m + 8 n - 2) F(j + 1, n0) + (2 j + 4 m - 8 n + 2) F(j + 2, n0) = G(j, n0 + 1) - G(j, n0), (Check!) and the theorem follows upon summing with respect to, n0, . QED. For the variable j, and F2, we have the following lemma A PROOF OF A RECURRENCE By Shalosh B. Ekhad, Rutgers University Theorem: Let, F(j, k), be given by (n + m) k (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k) and let, SUM(j), be the sum of, F(j, k), with respect to, k SUM(j), satisfies the following linear recurrence equation (j - 3 n) SUM(j) + (-3 j - 4 m + 8 n - 2) SUM(j + 1) + (2 j + 4 m - 8 n + 2) SUM(j + 2) =0. PROOF: We cleverly construct, G(j, k), := (n + m) k (n + m + k) (-2 n + m + j - k) (-1) (-1) binomial(2 n + 2 m, n + m + k) binomial(4 n - 2 m - j, 2 n - m - k)/(-4 n + 2 m + j) with the motive that (j - 3 n) F(j, k) + (-3 j - 4 m + 8 n - 2) F(j + 1, k) + (2 j + 4 m - 8 n + 2) F(j + 2, k) = G(j, k + 1) - G(j, k), (Check!) and the theorem follows upon summing with respect to, k, . QED. One can see that these are recurrences are identical -------------------------------------------- This finishes this article that took, 6.166, seconds.