A389060 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(3*k,n-2*k).

1, 0, 2, 9, 18, 65, 245, 735, 2506, 8784, 29277, 100320, 347589, 1193049, 4123821, 14318759, 49682490, 172830228, 602551604, 2102511006, 7346864653, 25706596155, 90032376546, 315635979373, 1107596435917, 3889779452940, 13671032884875, 48082756125105

Offset

0,3

Links

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

Formula

a(n) = [x^n] (1 + x^2 * (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^2 * (1 + x)^3) ). See A389063.

Mathematica

Table[Sum[Binomial[n, k]*Binomial[3*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Sep 25 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(3*k, n-2*k));
(Magma) [&+[Binomial(n, k)*Binomial(3*k, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // Vincenzo Librandi, Sep 25 2025

Crossrefs

Cf. A378406, A389063.

Sequence in context: A083708 A280588 A191520 * A037421 A083423 A068978

Adjacent sequences: A389057 A389058 A389059 * A389061 A389062 A389063

Keyword

nonn

Author

Seiichi Manyama, Sep 22 2025

Status

approved