A389290 - OEIS

A389290

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

1, 0, 0, 3, 4, 0, 21, 56, 36, 165, 660, 858, 1729, 7280, 14280, 23868, 81396, 203490, 364287, 975821, 2699004, 5492630, 12728430, 35036820, 79616745, 176181525, 459439344, 1117259220, 2500176536, 6173109360, 15409747716, 35569139848, 84964295988, 211754020569

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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 - 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^3 - x^4) ). See A365725.

MATHEMATICA

Table[Sum[Binomial[n+k-1, k]Binomial[k, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 06 2025 *)

PROG

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