A387617 - OEIS

A387617

a(n) = Sum_{k=0..floor(n/2)} binomial(n+k-1,k) * binomial(n+3*k-1,n-2*k).

1, 0, 2, 15, 70, 325, 1631, 8302, 42150, 215160, 1105577, 5705128, 29535883, 153363665, 798418070, 4165950640, 21779722550, 114064347585, 598308942080, 3142745677180, 16528816279745, 87030611968630, 458728613733612, 2420217998986450, 12780095256135675, 67540569663705450

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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/(1 - x^2 / (1 - x)^5)^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)^5) ). See A389347.

MATHEMATICA

Table[Sum[Binomial[n+k-1, k]Binomial[n+3*k-1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Oct 07 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\2, binomial(n+k-1, k)*binomial(n+3*k-1, n-2*k));