A387006 G.f. A(x) satisfies A(x) = 1 + x/(1+x^3) * A(x)^2.

1, 1, 2, 5, 13, 38, 117, 374, 1226, 4100, 13932, 47968, 166976, 586673, 2077814, 7410174, 26588203, 95913834, 347655258, 1265538557, 4624629016, 16958809018, 62386998767, 230172892750, 851474002794, 3157576995877, 11736012517772, 43711823069399

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))).

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

Mathematica

Table[Sum[ (-1)^k*Binomial[n-2*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(n-2*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
(Magma) [&+[Catalan(n-3*k) * (-1)^k*Binomial(n-2*k-1, k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025

Crossrefs

Cf. A000108, A376574.

Sequence in context: A149857 A001475 A360271 * A343937 A369729 A149858

Adjacent sequences: A387003 A387004 A387005 * A387007 A387008 A387009

Keyword

nonn

Author

Seiichi Manyama, Oct 26 2025

Status

approved