A068767 Generalized Catalan numbers.

1, 1, 10, 105, 1150, 13050, 152500, 1825625, 22293750, 276758750, 3483287500, 44352006250, 570333187500, 7396680812500, 96638930625000, 1270796364765625, 16806545339843750, 223400240246093750

Offset

0,3

Comments

a(n) = K(5,5; n)/5 with K(a,b; n) defined in a comment to A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..860

Formula

a(n) = (5^n) * p(n, -4/5) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 5*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-20*x*(1-4*x)))/(10*x).

(n+1)*a(n) = 80*(2-n)*a(n-2) + 10*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(10+10*sqrt(5)) * (10+2*sqrt(5))^n / (10*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

Equivalently, a(n) ~ 2^(2*n) * 5^((n-1)/2) * phi^(n + 1/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021

Mathematica

CoefficientList[Series[(1-Sqrt[1-20*x*(1-4*x)])/(10*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Crossrefs

Cf. A000108, A068764-A068766, A068768-A068772, A025227-A025230.

Sequence in context: A190955 A004343 A163166 * A233830 A046715 A145713

Adjacent sequences: A068764 A068765 A068766 * A068768 A068769 A068770

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved