A068771 Generalized Catalan numbers 9*x*A(x)^2 -A(x) +1 -8*x=0.

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,changed

Author

Wolfdieter Lang, Mar 04 2002

Status

approved