A389326 - OEIS

A389326

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(n+2*k-1,n-2*k).

1, 0, 2, 12, 46, 180, 770, 3332, 14350, 62184, 271482, 1190420, 5236726, 23107240, 102238292, 453396972, 2014720590, 8968626352, 39987878414, 178545845364, 798231606846, 3572840398160, 16008789864564, 71799930949428, 322311294528790, 1448046325552480

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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)^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^2 / (1 - x)^4) ). See A389247.

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[n+2*k-1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 08 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(n+2*k-1, n-2*k));