A023433 - OEIS

%I #27 Feb 03 2025 09:36:30

%S 1,1,1,1,1,2,4,7,12,21,38,70,129,238,442,827,1556,2939,5570,10593,

%T 20214,38690,74251,142844,275430,532215,1030440,1998733,3883552,

%U 7557865,14730670,28751455,56192036,109959882,215431019,422541192,829642870,1630613418

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

%H Alois P. Heinz, <a href="/A023433/b023433.txt">Table of n, a(n) for n = 0..1000</a>

%F 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

%F 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

%F G.f. A(x) satisfies: A(x) = (1 + x^3 * A(x)^2) / (1 - x + x^3 + x^4). - _Ilya Gutkovskiy_, Jul 20 2021

%p a:= proc(n) option remember;

%p       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-3-k), k=2..n-3))

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, May 08 2011

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

%Y Cf. A000108, A001006, A004148, A006318.

%K nonn,easy,changed

%O 0,6

%A _Olivier Gérard_