A023428 Generalized Catalan Numbers x^4*A(x)^2 -(1-x+x^4+x^5)*A(x) +1 =0.

1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 45, 77, 132, 224, 378, 640, 1093, 1881, 3250, 5622, 9732, 16874, 29332, 51126, 89313, 156283, 273842, 480474, 844220, 1485472, 2617335, 4617243, 8154289, 14415869, 25511256, 45190366, 80124434, 142189496

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Crossrefs

Cf. A000108, A001006, A004148, A006318.

Programs

Maple

A023428 := proc(n)
    option remember;
    if n = 0 then
        1 ;
    else
        procname(n-1)+add(procname(k)*procname(n-4-k),k=2..n-4) ;
    end if;
end proc:
seq(A023428(n),n=0..80) ; # R. J. Mathar, Oct 31 2014

Mathematica

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 2, n-4} ];

Formula

a(0)=1; a(n) = a(n-1) + Sum_{k=2..n-4} a(k)*a(n-4-k).

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x + x^4 + x^5). - Ilya Gutkovskiy, Jul 20 2021

Extensions

More terms from Sean A. Irvine, Jun 04 2019