A068771 Generalized Catalan numbers.

1, 1, 18, 333, 6318, 122634, 2429028, 48974949, 1002875094, 20814628158, 437088964860, 9272342710962, 198456435657036, 4280758166952756, 92972201833888200, 2031520673763657621, 44630859892110807654

Offset

0,3

Links

Fung Lam, Table of n, a(n) for n = 0..725

Formula

a(n) = (9^n) * p(n, -8/9) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 9*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-36*x*(1-8*x)))/(18*x).

Recurrence: (n+1)*a(n) = 288*(2-n)*a(n-2) + 18*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(2) * 24^n / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

Mathematica

a[n_] := (288 (2 - n) a[n - 2] + 18 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}](* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-36*x*(1-8*x)])/(18*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Crossrefs

Cf. A000108, A068764-A068770, A068772, A025227-A025230.

Sequence in context: A222812 A166787 A280806 * A039646 A212669 A158590

Adjacent sequences: A068768 A068769 A068770 * A068772 A068773 A068774

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved