A370623 Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^2) )^(3*n).

1, 0, 6, 9, 90, 255, 1671, 6258, 34674, 148455, 765141, 3499551, 17487531, 82704921, 408192420, 1964826174, 9657348546, 46944246777, 230604062127, 1127574041325, 5543828629305, 27211172907207, 133970970691311, 659351846223252, 3251029812112995

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(n-k-1,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-x^2)^3 / (1-x)^3 ). See A369013.

Mathematica

a[n_]:=SeriesCoefficient[((1-x)/(1-x-x^2))^(3n), {x, 0, n}]; Array[a, 25, 0] (* Stefano Spezia, May 01 2024 *)

Prog

(PARI) a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));

Crossrefs

Cf. A370620, A370621, A370622.

Cf. A369013.

Sequence in context: A372412 A377680 A103107 * A377683 A121233 A156180

Adjacent sequences: A370620 A370621 A370622 * A370624 A370625 A370626

Keyword

nonn

Author

Seiichi Manyama, May 01 2024

Status

approved