A091695 Expansion of e.g.f. exp(x/(1-x)^3).

1, 1, 7, 55, 529, 6121, 82711, 1273567, 21945505, 417540529, 8680953511, 195582295591, 4742407056817, 123045795823705, 3399348471640759, 99573135106176271, 3081061456572152641, 100382623544966098657, 3433727597233037475655, 123000248740384210119319, 4603377404407810366309201

Offset: 0

Author

Karol A. Penson, Jan 29 2004

Keywords

• nonn

Comments

Special values of the hypergeometric function 3F3: a(n) = n!*binomial(n+1,n-1) * hypergeom([ -n+1, 1/2*n+1, 1/2*n+3/2], [4/3, 5/3, 2], -4/27) for n>0.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..422

Crossrefs

Column k=3 of A293012.

Cf. A082579.

Programs

Mathematica

CoefficientList[Series[E^(x/(1-x)^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)

PARI

x='x+O('x^33);
Vec(serlaplace(exp( x/(1-x)^3 )))
/* Joerg Arndt, Sep 14 2012 */

Formula

E.g.f.: exp(x/(1-x)^3).

a(n) ~ 1/2*exp(-1/27-n^(1/4)*3^(3/4)/72+sqrt(3*n)/6+4/3*n^(3/4)*3^(1/4)-n)*3^(1/8)*n^(n-1/8). - Vaclav Kotesovec, Jun 27 2013

a(n) = n! * Sum_{k=0..n} binomial(n+2*k-1,n-k)/k!. - Seiichi Manyama, Mar 06 2023

Extensions

Prepended a(0)=1, Joerg Arndt, Sep 14 2012.