A389153 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+3*k,n-2*k).

1, 1, 3, 19, 91, 391, 1767, 8331, 39411, 186427, 887113, 4246265, 20400931, 98295211, 474876377, 2299723429, 11160126595, 54255979843, 264196953489, 1288363469313, 6290949805081, 30754387169133, 150509757577149, 737307535599517, 3615128526456027

Offset

0,3

Comments

Binomial transform of A389151.

Links

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

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389151(k).

a(n) = [x^n] (1 + x + x^2 * (1 + x)^4)^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 + x^2 * (1 + x)^4) ). See A389157.

Mathematica

Table[Sum[Binomial[n, k]*Binomial[n+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(n+3*k, n-2*k));
(Magma) [&+[Binomial(n, k)*Binomial(n+3*k, n-2*k): k in [0..Floor(n/2)]]: n in [0..30]]; // Vincenzo Librandi, Sep 25 2025

Crossrefs

Cf. A389157.

Sequence in context: A183384 A050863 A376836 * A283380 A049153 A074361

Adjacent sequences: A389150 A389151 A389152 * A389154 A389155 A389156

Keyword

nonn

Author

Seiichi Manyama, Sep 25 2025

Status

approved