A370620 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(3*n).

1, 3, 27, 255, 2535, 25908, 269667, 2843214, 30264975, 324543495, 3500669172, 37940361660, 412830243735, 4507040972190, 49345845670470, 541602648192480, 5957253066586815, 65650003858745514, 724693081872783375, 8011727857439155500, 88692087094226151300

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(4*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 ). See A368963.

Mathematica

a[n_]:=SeriesCoefficient[(1-x-x^2)^(-3*n), {x, 0, n}]; Array[a, 21, 0] (* Stefano Spezia, May 01 2024 *)

Prog

(PARI) a(n, s=2, t=3, u=0) = 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. A370621, A370622, A370623.

Cf. A368963.

Sequence in context: A235373 A361895 A279658 * A373818 A378612 A026294

Adjacent sequences: A370617 A370618 A370619 * A370621 A370622 A370623

Keyword

nonn

Author

Seiichi Manyama, May 01 2024

Status

approved