A111994 Sixth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

1, 6, 33, 176, 930, 4908, 25954, 137712, 733539, 3922834, 21060099, 113481504, 613619332, 3328768344, 18112655748, 98833261600, 540705999621, 2965360687518, 16299708148901, 89784615643728, 495545294427558

Offset: 0

Author

Wolfdieter Lang, Sep 12 2005

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. Sixth column of convolution triangle A011117.

Programs

Mathematica

CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^6, {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))^6) \\ G. C. Greubel, Mar 16 2017

Formula

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

a(n)= (6/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+5,k-1).

a(n) = 6*hypergeom([1-n, n+7], [2], -1), n>=1, a(0)=1.

Recurrence: n*(n+6)*a(n) = (7*n^2+30*n+5)*a(n-1) - (7*n^2+12*n-22)*a(n-2) + (n-3)*(n+3)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 3*sqrt(3*sqrt(2)-4)*(58-41*sqrt(2)) * (3+2*sqrt(2))^(n+6)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012