A389874 G.f. A(x) satisfies A(x) = 1 + x*(1+x^3)^3*A(x)^2.

1, 1, 2, 5, 17, 54, 177, 600, 2090, 7418, 26730, 97556, 359902, 1339950, 5028198, 18998181, 72214530, 275961940, 1059580185, 4085690663, 15814755274, 61428318546, 239358417367, 935373031350, 3664997706582, 14395415002272, 56670307154998, 223560965020304

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^3)^3)).

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

Mathematica

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

Crossrefs

Cf. A000108, A360272, A389873.

Sequence in context: A263678 A133510 A191640 * A390104 A148408 A002692

Adjacent sequences: A389871 A389872 A389873 * A389875 A389876 A389877

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 18 2025

Status

approved