A157074 - OEIS

%I #11 Jan 28 2022 07:40:07

%S 2,150,6252,182500,4112502,75578370,1173777752,15795816120,

%T 187652162502,1996568642530,19245807386652,169668375420180,

%U 1378768046330402,10396793993805030,73166155146412752,482928212647720720,3002693915693248002,17655197338344400470

%N Number of integer sequences of length n+1 with sum zero and sum of absolute values 50.

%H T. D. Noe, <a href="/A157074/b157074.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_51">Index entries for linear recurrences with constant coefficients</a>, 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).

%F 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).

%F From _G. C. Greubel_, Jan 27 2022: (Start)

%F a(n) = (n+1)*binomial(n+24, 25)*Hypergeometric3F2([-24, -n, 1-n], [2, -n-24], 1).

%F 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).

%F 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)

%t A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];

%t A157074[n_]:= A103881[n, 25];

%t Table[A157074[n], {n, 50}] (* _G. C. Greubel_, Jan 27 2022 *)

%o (Sage)

%o 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) )

%o def A157074(n): return A103881(n, 25)

%o [A157074(n) for n in (1..50)] # _G. C. Greubel_, Jan 27 2022

%Y Cf. A103881, A156554.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 22 2009