login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A157074
Number of integer sequences of length n+1 with sum zero and sum of absolute values 50.
2
2, 150, 6252, 182500, 4112502, 75578370, 1173777752, 15795816120, 187652162502, 1996568642530, 19245807386652, 169668375420180, 1378768046330402, 10396793993805030, 73166155146412752, 482928212647720720, 3002693915693248002, 17655197338344400470
OFFSET
1,1
LINKS
Index entries for linear recurrences with constant coefficients, signature (51,-1275,20825,-249900,2349060,-18009460, 115775100,-636763050,3042312350,-12777711870,47626016970,-158753389900, 476260169700,-1292706174900,3188675231420,-7174519270695,14771069086725, -27900908274925,48459472266975,-77535155627160,114456658306760, -156077261327400,196793068630200,-229591913401900,247959266474052, -247959266474052,229591913401900,-196793068630200,156077261327400, -114456658306760,77535155627160,-48459472266975,27900908274925,-14771069086725, 7174519270695,-3188675231420,1292706174900,-476260169700,158753389900, -47626016970,12777711870,-3042312350,636763050,-115775100,18009460,-2349060, 249900,-20825,1275,-51,1).
FORMULA
a(n) = T(n,25); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
From G. C. Greubel, Jan 27 2022: (Start)
a(n) = (n+1)*binomial(n+24, 25)*Hypergeometric3F2([-24, -n, 1-n], [2, -n-24], 1).
a(n) = (126410606437752/50!)*n*(n+1)*(9623905480333281923493425053824177930240000000000 + 27100515339271296805042905104567762524569600000000*n + 49226599934719560481828455826236675352166400000000*n^2 + 51923175705445481350794593882064923048017920000000*n^3 + 46502829595021715716879102923565907828539392000000*n^4 + 28607394119885617552139740430561122618473185280000*n^5 + 16559588497213417883781098164439679738807582720000*n^6 + 6903192311627666498917104674104501458397298688000*n^7 + 2894036204442771597885580471785456670461945446400*n^8 + 882529358789488763775646630321568918645729918976*n^9 + 285704714285545970609012303782721701384304001024*n^10 + 66744193695557588078616189319402098781536485376*n^11 + 17394219679949413313652735722550417627568410624*n^12 + 3209212575849629078911083109861120504852463616*n^13 + 693340015644326307061765976396831207893776384*n^14 + 103183723405307213352941409893689330849622016*n^15 + 18890270959451165193941203482138711306505984*n^16 + 2301923694341735297363581288294981193895936*n^17 + 363246399568340082151669298560235347864064*n^18 + 36632957463825141955678003229613126558336*n^19 + 5051271387716061681982819535517710183664*n^20 + 424699960734096109443243714664325553216*n^21 + 51748891662219811557282274201341501784*n^22 + 3644289612230496197746802122023398616*n^23 + 396093870596357042648294916274601009*n^24 + 23416970176809393473086005534732576*n^25 + 2288479608700865971390942858179924*n^26 + 113584302206510356395975946196976*n^27 + 10049618631034902174836327665474*n^28 + 417815336521106587249172637024*n^29 + 33668912037122043295220280476*n^30 + 1166960621063436872100315624*n^31 + 86099204270791153803452751*n^32 + 2468552637980851499947584*n^33 + 167536461123588897837416*n^34 + 3927535896285273089184*n^35 + 246218365513296690316*n^36 + 4640273089678232064*n^37 + 269714108783157936*n^38 + 3986042964314664*n^39 + 215542329647711*n^40 + 2405227111584*n^41 + 121370670916*n^42 + 961331184*n^43 + 45396066*n^44 + 227424*n^45 + 10076*n^46 + 24*n^47 + n^48).
G.f.: 2*x*(1 + 24*x + 576*x^2 + 6624*x^3 + 76176*x^4 + 558624*x^5 + 4096576*x^6 + 21507024*x^7 + 112911876*x^8 + 451647504*x^9 + 1806590016*x^10 + 5720868384*x^11 + 18116083216*x^12 + 46584213984*x^13 + 119787978816*x^14 + 254549454984*x^15 + 540917591841*x^16 + 961631274384*x^17 + 1709566710016*x^18 + 2564350065024*x^19 + 3846525097536*x^20 + 4895577396864*x^21 + 6230734868736*x^22 + 6749962774464*x^23 + 7312459672336*x^24 + 6749962774464*x^25 + 6230734868736*x^26 + 4895577396864*x^27 + 3846525097536*x^28 + 2564350065024*x^29 + 1709566710016*x^30 + 961631274384*x^31 + 540917591841*x^32 + 254549454984*x^33 + 119787978816*x^34 + 46584213984*x^35 + 18116083216*x^36 + 5720868384*x^37 + 1806590016*x^38 + 451647504*x^39 + 112911876*x^40 + 21507024*x^41 + 4096576*x^42 + 558624*x^43 + 76176*x^44 + 6624*x^45 + 576*x^46 + 24*x^47 + x^48)/(1-x)^51. (End)
MATHEMATICA
A103881[n_, k_]:= (n+1)*Binomial[n+k-1, k]*HypergeometricPFQ[{1-n, -n, 1-k}, {2, 1-n - k}, 1];
A157074[n_]:= A103881[n, 25];
Table[A157074[n], {n, 50}] (* G. C. Greubel, Jan 27 2022 *)
PROG
(Sage)
def A103881(n, k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157074(n): return A103881(n, 25)
[A157074(n) for n in (1..50)] # G. C. Greubel, Jan 27 2022
CROSSREFS
Sequence in context: A209077 A141139 A141130 * A329712 A128350 A046473
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 22 2009
STATUS
approved