A052885 E.g.f. A(x) is inverse to F(x) = x*exp(-x)/(1+x).

0, 1, 4, 33, 424, 7445, 166176, 4505053, 143787904, 5282091081, 219531404800, 10184792907641, 521761503753216, 29254578504622237, 1781920872844693504, 117169936148978011125, 8272258025961978167296

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..357

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 858

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

I. Mezo, A. Baricz, On the generalization of the Lambert W function with applications in theoretical physics, arXiv preprint arXiv:1408.3999 [math.CA], 2014-2015.

Formula

E.g.f.: RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z).

E.g.f. A(x) = sum(n>0, a(n)*x^n/n!) is inverse to F(x)=x*exp(-x)/(1+x), a(n)=(n-1)!*sum_{i=0..n-1} (n^(n-i-1)*binomial(n,i))/(n-i-1)!, n>0. - Vladimir Kruchinin, Jan 31 2012

a(n) ~ 5^(-1/4) * ((3+sqrt(5))/2)^n * exp((sqrt(5)-3)*n/2) * n^(n-1). - Vaclav Kotesovec, Jan 23 2014

Maple

spec := [S, {B=Prod(Z, C), C=Set(S), S=Sequence(B, 1<= card)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

CoefficientList[InverseSeries[Series[x/(E^x*(1+x)), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2014 *)

Prog

(Maxima) a(n):=((n-1)!*sum((n^(n-i-1)*binomial(n, i))/(n-i-1)!, i, 0, n-1)); /* Vladimir Kruchinin, Jan 31 2012 */

Crossrefs

Sequence in context: A111534 A162655 A216135 * A277184 A192548 A119821

Adjacent sequences: A052882 A052883 A052884 * A052886 A052887 A052888

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved