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

1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 65, 133, 274, 568, 1184, 2481, 5223, 11042, 23434, 49908, 106633, 228505, 490999, 1057683, 2283701, 4941502, 10713941, 23272929, 50642017, 110377543, 240944076, 526717211, 1152996206, 2527166334, 5545804784, 12184053993

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2795

Crossrefs

Cf. A000108, A001006, A004148, A004149, A023422, A023423.

Fourth row of A064645.

Programs

Maple

A023421 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        procname(n-1)+add(procname(k)*procname(n-2-k), k=3..n-2) ;
    end if;
end proc: # R. J. Mathar, May 01 2015

Mathematica

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 *)

PARI

{a(n) = if(n==0,1, a(n-1) + sum(k=3,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). - 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) +3*(n-7)*a(n-6) +(2*n-17)*a(n-7) +(n-10)*a(n-8)=0. - R. J. Mathar, Feb 03 2025