A387358 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*k,n-k).

1, 1, 7, 28, 131, 621, 2986, 14568, 71667, 355159, 1770007, 8862217, 44543786, 224619864, 1135853824, 5757673383, 29247575203, 148847761323, 758776529977, 3873722497342, 19802568285331, 101353224099561, 519312615801679, 2663507435418221, 13673395654874730

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] (1 + x * (1 + x)^3)^n.

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)^3) ). See A364742.

a(n) ~ sqrt(4 + sqrt(3) + 12*sqrt(3/208 + 5/(26*sqrt(3)))) * (1/2 + sqrt(3) + sqrt(17 + 100/(3*sqrt(3)))/2)^n / (2*sqrt(6*Pi*n)). - Vaclav Kotesovec, Oct 19 2025

Mathematica

Table[Sum[Binomial[n, k]Binomial[3*k, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(3*k, n-k));
(Magma) [&+[Binomial(n, k) * Binomial(3*k, n-k) : k in [0..n] ]: n in [0..40]]; // Vincenzo Librandi, Oct 08 2025

Crossrefs

Cf. A002426, A385975, A389225.

Cf. A364742, A378403.

Sequence in context: A219737 A316106 A359203 * A354456 A249872 A238448

Adjacent sequences: A387355 A387356 A387357 * A387359 A387360 A387361

Keyword

nonn

Author

Seiichi Manyama, Sep 25 2025

Status

approved