%I #30 May 01 2024 09:01:25

%S 1,3,19,132,963,7228,55264,428067,3347843,26378079,209065644,

%T 1664967747,13312423056,106798422942,859244421187,6930167382832,

%U 56015610380931,453628706358333,3679805451367471,29895358350622638,243204082036270588,1980931117038586824

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

%H Seiichi Manyama, <a href="/A370159/b370159.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*n-k,n-2*k).

%F The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ). See A369477.

%t a[n_]:=SeriesCoefficient[((1+x)*(1+x+x^2)^2)^n,{x,0,n}]; Array[a,22,0] (* _Stefano Spezia_, Apr 30 2024 *)

%o (PARI) a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

%Y Cf. A027908, A370160.

%Y Cf. A369477, A370194.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Feb 11 2024