A068772 Generalized Catalan numbers 10*x*A(x)^2 -A(x) +1 -9*x =0.

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,changed

Author

Wolfdieter Lang, Mar 04 2002

Status

approved