A023422 Generalized Catalan Numbers.

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 129, 261, 530, 1080, 2208, 4528, 9313, 19207, 39714, 82314, 170996, 355976, 742545, 1551817, 3248823, 6812947, 14309557, 30099645, 63402315

Offset

0,7

Links

Seiichi Manyama, Table of n, a(n) for n = 0..2926

A. Goupil, M.-E. Pellerin and J. de Wouters d'oplinter, Snake Polyominoes, arXiv preprint arXiv:1307.8432 [math.CO], 2013-2014. (Gives a g.f.)

Formula

G.f. A(x) satisfies: A(x) = (1 + x^2 * A(x)^2) / (1 - x + x^2 + x^3 + x^4 + x^5). - Ilya Gutkovskiy, Jul 20 2021

Mathematica

a[0]=1; a[n_]:= a[n]=a[n-1] + Sum[a[k]*a[n-2-k], {k, 4, n-2}]; Table[a[n], {n, 0, 30}] (* modified by G. C. Greubel, Jan 01 2018 *)
B[q_] = (q^2 + q^3 + q^4 + q^5 - Sqrt[((q(q^5 - 1))/(q - 1) - 1)^2 - 4q^6] - q + 1)/(2q^2); CoefficientList[B[q] + O[q]^31, q] (* Jean-François Alcover, Jan 29 2019 *)

Prog

(PARI) {a(n) = if(n==0, 1, a(n-1) + sum(k=4, n-2, a(k)*a(n-k-2)))};
for(n=0, 30, print1(a(n), ", ")) \\ G. C. Greubel, Jan 01 2018

Crossrefs

Cf. A000108, A001006, A004148, A004149, A023421, A023423.

Fifth row of A064645.

Sequence in context: A278995 A117302 A265407 * A084638 A157021 A210543

Adjacent sequences: A023419 A023420 A023421 * A023423 A023424 A023425

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved