A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2*k,2*n-2*k).
1, 8, 44, 336, 2166, 11832, 60576, 292248, 1334817, 5840296, 24637976, 100684376, 400255050, 1553016960, 5897388492, 21967711160, 80425346844, 289868771928, 1029979010972, 3612517052608, 12520285820362, 42919328903928, 145643017892472, 489606988741128
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Index entries for linear recurrences with constant coefficients, signature (16, -104, 352, -708, 1232, -2800, 5168, -6122, 9728, -18008, 16816, -19152, 37744, -27096, 24960, -50611, 24960, -27096, 37744, -19152, 16816, -18008, 9728, -6122, 5168, -2800, 1232, -708, 352, -104, 16, -1).
Programs
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Magma
[&+[Binomial(k+7, 7) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025
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Mathematica
Table[Sum[Binomial[k+7,7]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *) -
PARI
a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(2*k, 2*n-2*k));
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PARI
my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))
Formula
G.f.: (Sum_{k=0..4} 4^k * binomial(8,2*k) * (1-x-x^2)^(8-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^8.