A389375 - OEIS

A389375

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

1, 0, 0, 3, 8, 15, 45, 147, 408, 1128, 3325, 9801, 28421, 82953, 244293, 719583, 2119512, 6259485, 18525060, 54879980, 162758613, 483316683, 1436757839, 4274856843, 12730177173, 37940570190, 113160456435, 337738702323, 1008660020061, 3014174680521, 9012255231145

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OFFSET

0,4

LINKS

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

FORMULA

a(n) = [x^n] 1/(1 - x^3 / (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^3 / (1 - x)^2) ). See A389348.

MATHEMATICA

Table[Sum[Binomial[n+k-1, k]Binomial[n-k-1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 04 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(n+k-1, k)*binomial(n-k-1, n-3*k));