A390135 G.f. A(x) satisfies A(x) = 1 + x/(1+x^3)^3 * A(x)^2.

1, 1, 2, 5, 11, 30, 87, 267, 842, 2711, 8869, 29394, 98485, 333038, 1135194, 3896244, 13454040, 46707272, 162924504, 570746300, 2007110128, 7082934600, 25074540508, 89024971896, 316917263448, 1130950454685, 4045049192534, 14498211521126, 52065611173173, 187316296483994

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)^3)).

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3*n-8*k-1,k) * Catalan(n-3*k).

Mathematica

Table[Sum[ (-1)^k*Binomial[3*n-8*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(3*n-8*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
(Magma) [&+[Catalan(n-3*k) * (-1)^k*Binomial(3*n-8*k-1, k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025

Crossrefs

Cf. A000108, A390104.

Sequence in context: A292210 A079225 A389895 * A139466 A291139 A139467

Adjacent sequences: A390132 A390133 A390134 * A390136 A390137 A390138

Keyword

nonn

Author

Seiichi Manyama, Oct 26 2025

Status

approved