A389322 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+3*k-1,n-k).

1, 1, 9, 55, 345, 2236, 14727, 98134, 659801, 4467160, 30413924, 208016337, 1428159207, 9836845630, 67941521286, 470386502500, 3263507004569, 22683873869360, 157929616709400, 1101154915382352, 7687901741554980, 53738680702979509, 376043976132232153

Offset

0,3

Links

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

Formula

a(n) = [x^n] (1 + x / (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 / (1 - x)^4) ). See A321798.

Mathematica

Table[Sum[Binomial[n, k]Binomial[n+3*k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 09 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+3*k-1, n-k));
(Magma) [&+[Binomial(n, k) * Binomial(n+3*k-1, n-k) : k in [0..n] ]: n in [0..40]]; // Vincenzo Librandi, Oct 09 2025

Crossrefs

Cf. A321798.

Sequence in context: A362365 A183805 A037578 * A096191 A362088 A281454

Adjacent sequences: A389319 A389320 A389321 * A389323 A389324 A389325

Keyword

nonn

Author

Seiichi Manyama, Sep 29 2025

Status

approved