A068769 Generalized Catalan numbers.

1, 1, 14, 203, 3038, 46746, 736764, 11853051, 194053622, 3224557406, 54265836548, 923218762270, 15854602773100, 274500192707860, 4786546243533432, 83989334625037947, 1481965556616225702

Offset

0,3

Comments

a(n) = K(7,7; n)/7 with K(a,b; n) defined in a comment to A068763.

Links

Fung Lam, Table of n, a(n) for n = 0..780

Formula

a(n) = (7^n) * p(n, -6/7) with the row polynomials p(n, x) defined from array A068763.

a(n+1) = 7*sum(a(k)*a(n-k), k=0..n), n>=1, a(0)=1=a(1).

G.f.: (1-sqrt(1-28*x*(1-6*x)))/(14*x).

Recurrence: (n+1)*a(n) = 168*(2-n)*a(n-2) + 14*(2*n-1)*a(n-1). - Fung Lam, Mar 04 2014

a(n) ~ sqrt(14+14*sqrt(7)) * (14+2*sqrt(7))^n / (14*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 04 2014

Mathematica

a[0] = 1; a[1] = 1; a[2] = 14; a[n_] := (168 (2 - n) a[n - 2] + 14 (2 n - 1) a[n - 1])/(n + 1); Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 04 2014 *)
CoefficientList[Series[(1-Sqrt[1-28*x*(1-6*x)])/(14*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 04 2014 *)

Crossrefs

Cf. A000108, A068764-A068768, A068770-A068772, A025227-A025230.

Sequence in context: A199707 A222586 A221830 * A113349 A109764 A349283

Adjacent sequences: A068766 A068767 A068768 * A068770 A068771 A068772

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 04 2002

Status

approved