A023422 Generalized Catalan Numbers x^2*A(x)^2 -(1-x+x^2+x^3+x^4+x^5)*A(x) + 1 =0.

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

Author

Olivier Gérard

Keywords

• nonn
• easy

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.)

Crossrefs

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

Fifth row of A064645.

Programs

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 *)

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

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

D-finite with recurrence (n+2)*a(n) +(-2*n-1)*a(n-1) +(-n+1)*a(n-2) +(n-4)*a(n-4) +(2*n-11)*a(n-5) +(n-7)*a(n-6) +2*(2*n-17)*a(n-7) +3*(n-10)*a(n-8) +(2*n-23)*a(n-9) +(n-13)*a(n-10)=0. - R. J. Mathar, Feb 03 2025