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%I #30 Jan 13 2024 10:44:37
%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.
%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
%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_
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