A088690 E.g.f.: A(x) = f(x*A(x)), where f(x) = (1+x)*exp(x).

1, 2, 11, 106, 1489, 27696, 643579, 17973488, 586899009, 21953140480, 925890264331, 43480125312768, 2250352192663249, 127280062346049536, 7811329076598534075, 517016126622623635456, 36713034605774835974401, 2784127167066690618458112

Offset: 0

Author

Paul D. Hanna, Oct 06 2003

Keywords

• nonn

Comments

Radius of convergence of A(x): r = tau^2*exp(-tau) = 0.20588... and A(r) = (1+tau)*exp(tau), where tau=(sqrt(5)-1)/2 and r = limit a(n)/a(n+1)*n as n->infinity.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Programs

Maple

a := n -> n!*simplify(hypergeom([-n], [2], -n-1)):
seq(a(n), n=0..15); # Peter Luschny, Apr 20 2016

Mathematica

CoefficientList[1/x*InverseSeries[Series[x*E^(-x)/(1+x), {x, 0, 21}], x],x]*Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2014 *)

PARI

a(n)=n!*polcoeff(((1+x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)

Formula

a(n) = n! * [x^n] ((1+x)*exp(x))^(n+1)/(n+1).

a(n) = Sum_{k=1..n} n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - Vladeta Jovovic, Jun 17 2006

E.g.f.: A(x) = (1/x)*series_reversion(x*exp(-x)/(1+x)). - Paul D. Hanna, Jun 17 2006

E.g.f.: B(x)/(1-x*B(x)), where B(x) is e.g.f. for A052873(). - Vladeta Jovovic, Jun 18 2006

a(n) ~ 5^(-1/4) * ((1+sqrt(5))/2)^(2*n+2) * exp((sqrt(5) - 1 - (3 - sqrt(5))*n)/2) * n^(n-1). - Vaclav Kotesovec, Jan 24 2014

a(n) = n!*hypergeom([-n], [2], -n-1). - Peter Luschny, Apr 20 2016