A088692 E.g.f: A(x) = f(x*A(x)), where f(x) = (1+2*x)*exp(x).

1, 3, 23, 304, 5829, 147696, 4670371, 177383424, 7874174601, 400298556160, 22940919680271, 1463679309053952, 102911522568495757, 7906731860604186624, 659108356837269579675, 59252790438687592677376, 5714517052927568389576209, 588555892122678050845556736

Offset: 0

Author

Paul D. Hanna, Oct 07 2003

Keywords

• nonn

Comments

Radius of convergence of A(x): r = (1/4)*exp(-1/2) = 0.151632.., where A(r) = 2*exp(1/2) and r = lim_{n->infinity} (a(n)/a(n+1))*n.

Links

• Robert Israel, Table of n, a(n) for n = 0..342

Crossrefs

Cf. A088690, A088693.

Programs

Maple

f := n -> simplify(exp(-(1/4)*n-1/4)*2^n*factorial(n)*((5*n+1)*WhittakerM(-n, 1/2, (1/2)*n+1/2)-(2*n-2)*WhittakerM(1-n, 1/2, (1/2)*n+1/2))/(n+1)^2):
map(f, [$0..30]); # Robert Israel, Oct 08 2017

Mathematica

Table[Sum[2^(n-k)*n^(k-2)*n!/k!*Binomial[n-1,k-1],{k,1,n}],{n,1,21}] (* Vaclav Kotesovec after Vladeta Jovovic, Jan 24 2014 *)

PARI

a(n)=n!*polcoeff(((1+2*x)*exp(x))^(n+1)+x*O(x^n),n,x)/(n+1)

Formula

a(n) = n! * [x^n] ((1+2*x)*exp(x))^(n+1)/(n+1).

a(n) = Sum_{k=1..n} 2^(n-k)*n^(k-2)*n!/k!*binomial(n-1,k-1) (offset 1). - Vladeta Jovovic, Jun 19 2006

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

a(n) = exp(-(n+1)/4)*2^n*n!*(n+1)^(-2)*((5*n+1)*WhittakerM(-n,1/2,(n+1)/2) - 2*(n-1)*WhittakerM(1-n,1/2,(n+1)/2)). - Robert Israel, Oct 08 2017