A068768 Generalized Catalan numbers.

1, 1, 12, 150, 1944, 25992, 356832, 5008824, 71629920, 1040509152, 15315578496, 227981324736, 3426473187072, 51929043390720, 792725911280640, 12178706839758720, 188158789025809920, 2921622674591946240

Offset

0,3

Comments

a(n) = K(6,6; n)/6 with K(a,b; n) defined in a comment to A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..810

Formula

a(n) = (6^n) * p(n, -5/6) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 6*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-24*x*(1-5*x)))/(12*x).

D-finite with recurrence: (n+1)*a(n) = 120*(2-n)*a(n-2) + 12*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(3+3*sqrt(6)) * (12+2*sqrt(6))^n / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

Mathematica

CoefficientList[Series[(1-Sqrt[1-24*x*(1-5*x)])/(12*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Crossrefs

Cf. A000108, A068764-A068767, A068769-A068772, A025227-A025230.

Sequence in context: A056351 A056345 A264233 * A053507 A060917 A113358

Adjacent sequences: A068765 A068766 A068767 * A068769 A068770 A068771

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved