A378407 a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(n+2*k,n-3*k).

1, 1, 1, 4, 25, 106, 352, 1114, 3865, 14539, 54886, 201763, 732568, 2679535, 9917818, 36903049, 137265337, 510201961, 1898730307, 7082472358, 26468394430, 99026247688, 370771000975, 1389387381691, 5211329801272, 19564292736706, 73504888190371, 276350941918741

Offset

0,4

Comments

Binomial transform of A389061. - Seiichi Manyama, Sep 22 2025

Links

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

Formula

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

a(n) = Sum_{k=0..n} binomial(n,k) * A389061(k). - Seiichi Manyama, Sep 22 2025

Mathematica

Table[Sum[Binomial[n, k]*Binomial[n+2*k, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 25 2025 *)

Prog

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

Crossrefs

Cf. A228960, A389061.

Sequence in context: A167889 A329495 A042651 * A225692 A359524 A070764

Adjacent sequences: A378404 A378405 A378406 * A378408 A378409 A378410

Keyword

nonn

Author

Seiichi Manyama, Nov 25 2024

Status

approved