A378153 - OEIS

A378153

G.f. A(x) satisfies A(x) = 1 + (x * (1+x))^3 * A(x)^2.

1, 0, 0, 1, 3, 3, 3, 12, 30, 45, 75, 192, 436, 798, 1554, 3542, 7740, 15543, 32183, 70794, 153252, 321431, 684123, 1491504, 3232672, 6928779, 14957787, 32615388, 70991040, 153985890, 335256886, 733206840, 1603258134, 3503385568, 7671749664, 16837946850

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OFFSET

0,5

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

FORMULA

a(n) = Sum_{k=0..floor(n/3)} binomial(3*k,n-3*k) * C(k), where C(k) are the Catalan numbers (A000108).

G.f.: 2/(1 + sqrt(1 - 4*(x*(1+x))^3)).

MATHEMATICA

Table[Sum[ Binomial[3*k, n-3*k]*CatalanNumber[k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)

PROG

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