A390134 - OEIS

A390134

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

1, 1, 2, 5, 12, 34, 102, 320, 1030, 3383, 11290, 38176, 130512, 450350, 1566458, 5486541, 19333804, 68496366, 243833190, 871719749, 3128496100, 11267070349, 40706748704, 147496902650, 535866079848, 1951617640107, 7123887216942, 26058587138333, 95506210663560

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OFFSET

0,3

LINKS

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

FORMULA

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

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-5*k-1,k) * Catalan(n-3*k).

MATHEMATICA

Table[Sum[ (-1)^k*Binomial[2*n-5*k-1, k]*CatalanNumber[n-3*k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-5*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));