A111996 Eighth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

1, 8, 52, 312, 1802, 10200, 57092, 317544, 1760035, 9738160, 53844184, 297717712, 1646893140, 9116815952, 50514367512, 280173703472, 1555632093093, 8647009926904, 48117998453036, 268057662257096, 1494927614877214

Offset

0,2

Links

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

Formula

G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^8.

a(n)= (8/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+7,k-1).

a(n) = 8*hypergeom([1-n, n+9], [2], -1), n>=1, a(0)=1.

Recurrence: n*(n+8)*a(n) = (7*n^2+44*n+21)*a(n-1) - (7*n^2+26*n-24)*a(n-2) + (n-3)*(n+5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ sqrt(3*sqrt(2)-4)*(338-239*sqrt(2)) * (3+2*sqrt(2))^(n+8)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Mathematica

CoefficientList[Series[((1+x-Sqrt[1-6x+x^2])/(4x))^8, {x, 0, 20}], x]  (* Harvey P. Dale, Apr 01 2011 *)

Prog

(PARI) x='x+O(x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^8) \\ G. C. Greubel, Mar 16 2017

Crossrefs

Cf. Eighth column of convolution triangle A011117.

Sequence in context: A279283 A257285 A125345 * A016129 A006631 A205218

Adjacent sequences: A111993 A111994 A111995 * A111997 A111998 A111999

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 12 2005

Status

approved