A023433 Generalized Catalan Numbers.

1, 1, 1, 1, 1, 2, 4, 7, 12, 21, 38, 70, 129, 238, 442, 827, 1556, 2939, 5570, 10593, 20214, 38690, 74251, 142844, 275430, 532215, 1030440, 1998733, 3883552, 7557865, 14730670, 28751455, 56192036, 109959882, 215431019, 422541192, 829642870, 1630613418

Offset

0,6

Links

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

Formula

Recurrence: (n+3)*a(n) = (2*n+3)*a(n-1) - n*a(n-2) + (2*n-3)*a(n-3) + (2*n-9)*a(n-5) - (n-6)*a(n-6) - (2*n-15)*a(n-7) - (n-9)*a(n-8). - Vaclav Kotesovec, Aug 25 2014

a(n) ~ c * d^n / n^(3/2), where d = 2.0423505898306085793498312456063... is the root of the equation -1 - 2*d - d^2 + d^3 - 2*d^4 + d^5 = 0, c = 1.36047848416839112694538628599558274531... . - Vaclav Kotesovec, Aug 25 2014

G.f. A(x) satisfies: A(x) = (1 + x^3 * A(x)^2) / (1 - x + x^3 + x^4). - Ilya Gutkovskiy, Jul 20 2021

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-3-k), k=2..n-3))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 08 2011

Mathematica

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-3-k ], {k, 2, n-3} ];

Crossrefs

Cf. A000108, A001006, A004148, A006318.

Sequence in context: A218600 A000709 A054161 * A190168 A288133 A005126

Adjacent sequences: A023430 A023431 A023432 * A023434 A023435 A023436

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved