A389376 - OEIS

A389376

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

1, 0, 0, 3, 12, 30, 81, 273, 924, 2937, 9270, 29997, 98121, 319995, 1042587, 3409113, 11184348, 36746214, 120852423, 398010195, 1312640802, 4334090274, 14324257167, 47385596337, 156892453737, 519884783805, 1723972146585, 5720690868465, 18995159088603

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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)^3)^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)^3) ). See A389349.

MATHEMATICA

Table[Sum[Binomial[n+k-1, k]Binomial[n-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-1, n-3*k));