A370160 - OEIS

A370160

Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^2)^2 )^n.

1, 4, 32, 286, 2688, 26004, 256322, 2559960, 25816576, 262307824, 2681024032, 27534988936, 283926200706, 2937573629800, 30480431060160, 317053438632786, 3305105501423616, 34519689280675808, 361146528603877520, 3784045825018539968, 39702608870540290688

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OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..970

FORMULA

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(4*n-k,n-2*k).

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^2) ). See A369478.

MATHEMATICA

a[n_]:=SeriesCoefficient[((1+x)^2*(1+x+x^2)^2)^n, {x, 0, n}]; Array[a, 21, 0] (* Stefano Spezia, Apr 30 2024 *)

PROG

(PARI) a(n, s=2, t=2, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));