A023429 Generalized Catalan Numbers x^4*A(x)^2 -(1-x+x^4+x^5+x^6)*A(x) + 1 =0.

1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 73, 122, 204, 340, 566, 945, 1586, 2674, 4521, 7656, 12982, 22047, 37509, 63934, 109166, 186685, 319679, 548091, 940819, 1616830, 2781706, 4790949, 8259748, 14253480, 24618561, 42557378

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A001006, A004148, A006318.

Programs

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-4-k), k=3..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 07 2011

Mathematica

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 3, n-4} ];

Formula

G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x + x^4 + x^5 + x^6). - Ilya Gutkovskiy, Jul 20 2021