A111997 Ninth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

1, 9, 63, 399, 2403, 14067, 80949, 460845, 2605590, 14666470, 82320714, 461238282, 2581644378, 14442658074, 80785970838, 451934259654, 2528977211775, 14157983986839, 79302044283297, 444448115168049, 2492468172937125

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Ninth column of convolution triangle A011117.

Programs

Mathematica

CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

PARI

x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^9) \\ G. C. Greubel, Mar 17 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^9.

a(n) = (9/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+8,k-1).

a(n) = 9*hypergeom([1-n, n+10], [2], -1), n>=1, a(0)=1.

Recurrence: n*(n+9)*a(n) = (7*n^2+51*n+32)*a(n-1) - (7*n^2+33*n-22)*a(n-2) + (n-3)*(n+6)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 9*sqrt(3*sqrt(2)-4)*(577-408*sqrt(2)) * (3+2*sqrt(2))^(n+9)/(64*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012