A370159 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^2 )^n.

1, 3, 19, 132, 963, 7228, 55264, 428067, 3347843, 26378079, 209065644, 1664967747, 13312423056, 106798422942, 859244421187, 6930167382832, 56015610380931, 453628706358333, 3679805451367471, 29895358350622638, 243204082036270588, 1980931117038586824

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(3*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) * (1+x+x^2)^2) ). See A369477.

Mathematica

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

Prog

(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));

Crossrefs

Cf. A027908, A370160.

Cf. A369477, A370194.

Sequence in context: A063395 A226459 A256092 * A369627 A074567 A332621

Adjacent sequences: A370156 A370157 A370158 * A370160 A370161 A370162

Keyword

nonn

Author

Seiichi Manyama, Feb 11 2024

Status

approved