A023423 Generalized Catalan Numbers x^2*A(x)^2 -(1-x+x^2+x^3+x^4+x^5+x^6)*A(x) + 1 =0.

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 257, 517, 1042, 2104, 4256, 8624, 17504, 35585, 72455, 147746, 301706, 616948, 1263240, 2589840, 5316033, 10924681, 22475831, 46290195

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3028

Crossrefs

Cf. A000108, A001006, A004148, A004149, A023421, A023422.

Sixth row of A064645.

Programs

Maple

A023423 := proc(n)
    option remember;
    if n <= 6 then
        1;
    else
        procname(n-1)+add(procname(k)*procname(n-2-k),k=5..n-2) ;
    end if;
end proc: # R. J. Mathar, Oct 10 2014

Mathematica

a[0]=1; a[n_]:= a[n]=a[n-1] + Sum[a[k]*a[n-2-k], {k,5,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=5,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 + x^6). - Ilya Gutkovskiy, Jul 20 2021