A389126 - OEIS

A389126

a(n) = Sum_{k=0..floor(n/4)} binomial(n,k) * binomial(2*k,n-4*k).

1, 0, 0, 0, 4, 10, 6, 0, 28, 144, 270, 220, 286, 1716, 5460, 9100, 10220, 23120, 86496, 217056, 354654, 538650, 1389850, 4108720, 9071062, 15513960, 29008980, 74809800, 192092940, 399254310, 734048406, 1540299852, 3828305196, 9018451552, 18422494992, 36278089760, 80237992912

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OFFSET

0,5

LINKS

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

FORMULA

a(n) = [x^n] (1 + x^4 * (1 + x)^2)^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^4 * (1 + x)^2) ). See A389130.

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[2*k, n-4*k], {k, 0, Floor[n/4]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 26 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\4, binomial(n, k)*binomial(2*k, n-4*k));