A389872 - OEIS

A389872

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

1, 1, 2, 8, 26, 90, 330, 1240, 4770, 18701, 74422, 299890, 1221226, 5017930, 20778462, 86622414, 363262562, 1531406342, 6486242016, 27587989081, 117785790794, 504612883661, 2168619710720, 9346567146703, 40388968308882, 174954656793291, 759557686502304, 3304438062381584

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OFFSET

0,3

LINKS

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

FORMULA

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

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

MATHEMATICA

Table[Sum[Binomial[3*(n-2*k), k]*CatalanNumber[n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 29 2025 *)

PROG

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