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A023421 Generalized Catalan Numbers x^2*A(x)^2 -(1-x+x^2+x^3+x^4)*A(x) + 1 =0.

%I A023421 #27 Feb 03 2025 09:35:43
%S A023421 1,1,1,1,1,2,4,8,16,32,65,133,274,568,1184,2481,5223,11042,23434,
%T A023421 49908,106633,228505,490999,1057683,2283701,4941502,10713941,23272929,
%U A023421 50642017,110377543,240944076,526717211,1152996206,2527166334,5545804784,12184053993
%N A023421 Generalized Catalan Numbers x^2*A(x)^2 -(1-x+x^2+x^3+x^4)*A(x) + 1 =0.
%H A023421 Seiichi Manyama, <a href="/A023421/b023421.txt">Table of n, a(n) for n = 0..2795</a>
%F A023421 G.f. A(x) satisfies: A(x) = (1 + x^2 * A(x)^2) / (1 - x + x^2 + x^3 + x^4). - _Ilya Gutkovskiy_, Jul 20 2021
%F A023421 D-finite with recurrence (n+2)*a(n) +(-2*n-1)*a(n-1) +(-n+1)*a(n-2) +(n-4)*a(n-4) +3*(n-7)*a(n-6) +(2*n-17)*a(n-7) +(n-10)*a(n-8)=0. - _R. J. Mathar_, Feb 03 2025
%p A023421 A023421 := proc(n)
%p A023421     option remember;
%p A023421     if n = 0 then
%p A023421         1;
%p A023421     else
%p A023421         procname(n-1)+add(procname(k)*procname(n-2-k), k=3..n-2) ;
%p A023421     end if;
%p A023421 end proc: # _R. J. Mathar_, May 01 2015
%t A023421 a[0]=1; a[n_]:= a[n]=a[n-1] + Sum[a[k]*a[n-2-k], {k,3,n-2}]; Table[a[n], {n,0,30}] (* modified by _G. C. Greubel_, Jan 01 2018 *)
%o A023421 (PARI) {a(n) = if(n==0,1, a(n-1) + sum(k=3,n-2, a(k)*a(n-k-2)))};
%o A023421 for(n=0,30, print1(a(n), ", ")) \\ _G. C. Greubel_, Jan 01 2018
%Y A023421 Cf. A000108, A001006, A004148, A004149, A023422, A023423.
%Y A023421 Fourth row of A064645.
%K A023421 nonn,easy
%O A023421 0,6
%A A023421 _Olivier Gérard_