A111995 Seventh convolution of Schroeder's (second problem) numbers A001003(n), n >= 0.

1, 7, 42, 238, 1316, 7196, 39158, 212738, 1155889, 6287015, 34249404, 186920468, 1022134288, 5600420336, 30745867316, 169116129308, 931937277257, 5144687596447, 28449040406262, 157571572143538, 874089046798212

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. Seventh column of convolution triangle A011117.

Programs

Mathematica

CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

PARI

my(x='x+O('x^50)); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^7) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^7.

a(n) = (7/n)*Sum_{k=1..n} binomial(n,k)*binomial(n+k+6,k-1).

a(n) = 7*hypergeom([1-n, n+8], [2], -1), n >= 1, a(0)=1.

a(n) = fourth binomial transform of 1,3,2,6,4,12. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Recurrence: n*(n+7)*a(n) = (7*n^2+37*n+12)*a(n-1) - (7*n^2+19*n-24)*a(n-2) + (n-3)*(n+4)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 7*sqrt(3*sqrt(2)-4)*(99-70*sqrt(2)) * (3+2*sqrt(2))^(n+7)/(32*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Extensions

Incorrect formula removed by Jason Yuen, Sep 07 2024