A023422 - OEIS

A023422

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

%I #34 Feb 03 2025 09:35:37

%S 1,1,1,1,1,1,2,4,8,16,32,64,129,261,530,1080,2208,4528,9313,19207,

%T 39714,82314,170996,355976,742545,1551817,3248823,6812947,14309557,

%U 30099645,63402315

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

%H Seiichi Manyama, <a href="/A023422/b023422.txt">Table of n, a(n) for n = 0..2926</a>

%H A. Goupil, M.-E. Pellerin and J. de Wouters d'oplinter, <a href="http://arxiv.org/abs/1307.8432">Snake Polyominoes</a>, arXiv preprint arXiv:1307.8432 [math.CO], 2013-2014. (Gives a g.f.)

%F 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

%F 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

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if(n==0,1, a(n-1) + sum(k=4,n-2, a(k)*a(n-k-2)))};

%o for(n=0,30, print1(a(n), ", ")) \\ _G. C. Greubel_, Jan 01 2018

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

%Y Fifth row of A064645.

%K nonn,easy

%O 0,7

%A _Olivier Gérard_