A068772 Generalized Catalan numbers.

1, 1, 20, 410, 8600, 184200, 4020000, 89205000, 2008700000, 45816140000, 1056825200000, 24618524200000, 578457724000000, 13695679012000000, 326448619920000000, 7827776361090000000, 188701194087000000000

Offset

0,3

Comments

This is the tenth member in the a-family of sequences K(a,a; n), a=1,2,3,...,n>=0, defined in a comment to the array A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..700

Formula

a(n) = (10^n) * p(n, -9/10) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 10*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-40*x*(1-9*x)))/(20*x).

Recurrence: (n+1)*a(n) = 360*(2-n)*a(n-2) + 20*(2*n-1)*a(n-1). - Fung Lam, Mar 05 2014

a(n) ~ sqrt(5+5*sqrt(10)) * (20+2*sqrt(10))^n / (10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014

Mathematica

a[0] = 1; a[1] = 1; a[n_] := (360 (2 - n) a[n - 2] + 20 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-40*x*(1-9*x)])/(20*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)

Crossrefs

Cf. A000108, A068764-A068771, A025227-A025230.

Sequence in context: A196740 A196898 A355966 * A230349 A158601 A268738

Adjacent sequences: A068769 A068770 A068771 * A068773 A068774 A068775

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved