A111993 Fifth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

1, 5, 25, 125, 630, 3206, 16470, 85350, 445775, 2344595, 12408903, 66042795, 353259900, 1898119100, 10240583420, 55454182716, 301307002605, 1642192132625, 8975693643525, 49186242980105, 270186765784210

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))^5.

a(n)= (5/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+4,k-1), a(0)=1.

a(n) = 5*hypergeom([1-n, n+6], [2], -1), n>=1, a(0)=1.

Recurrence: n*(n+5)*a(n) = n*(7*n+23)*a(n-1) - (n+2)*(7*n-9)*a(n-2) + (n-3)*(n+2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012

a(n) ~ 5*sqrt(3*sqrt(2)-4)*(17-12*sqrt(2)) * (3+2*sqrt(2))^(n+5)/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

Mathematica

CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^5, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)

Prog

(PARI) x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^5) \\ G. C. Greubel, Mar 16 2017

Crossrefs

Cf. Fifth column of convolution triangle A011117. Fourth convolution: A010849.

Sequence in context: A097680 A069030 A373281 * A341266 A113996 A340538

Adjacent sequences: A111990 A111991 A111992 * A111994 A111995 A111996

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 12 2005

Status

approved