A101054 E.g.f.: exp(exp(x)-1)/(1-x)^2.

1, 3, 12, 59, 343, 2302, 17537, 149711, 1417444, 14755947, 167664955, 2066153410, 27460121377, 391670868279, 5969233120520, 96827494235555, 1665874346755107, 30302732222472482, 581141491978752769, 11719848107849221435, 247960209086201040552, 5491993051193586857235

Offset: 0

Author

Karol A. Penson, Nov 29 2004

Keywords

• nonn

Comments

Sequence appears in the problem of normal ordering of functions of boson operators.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..447

Crossrefs

Cf. A000110, A101053.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(x)-1)/(1-x)^2)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 23 2018

Maple

egf := exp(exp(x)-1)/((1-x)^2);
seq(n!*coeff(series(egf,x,n+1),x,n),n=0..21); # Peter Luschny, Sep 17 2014

Mathematica

With[{nmax = 50}, CoefficientList[Series[Exp[Exp[x] - 1]/(1 - x)^2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, May 23 2018 *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(exp(x)-1)/(1-x)^2 )) \\ G. C. Greubel, May 23 2018

Formula

a(n) = ((-1)^n*n!/e)*Sum_{k>=0} L(n,-n-2,k)/k!, where L is a generalized Laguerre polynomial.

a(n) = Sum_{k=0..n} binomial(n,k)*(k + 1)!*Bell(n-k), where Bell() = A000110. - Ilya Gutkovskiy, May 24 2018

a(n) ~ exp(exp(1)-1) * n * n!. - Vaclav Kotesovec, Jun 26 2022