A136461 Expansion of e.g.f.: A(x) = -(1 + LambertW(-log(1+x))/log(1+x))/x.

1, 1, 3, 14, 96, 849, 9362, 123101, 1888016, 33066768, 651883152, 14286514186, 344690210928, 9079702374300, 259327537407416, 7983107543564724, 263518937698466304, 9285770278110061664, 347916420499685643072, 13812127364516107258944, 579183295530010157485824

Offset: 0

Author

Paul D. Hanna, Dec 31 2007

Keywords

• nonn

Comments

A033917 gives the coefficients of iterated exponential function defined by y(x) = x^y(x) expanded about x=1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 71 terms from Vincenzo Librandi)

Crossrefs

Cf. A033917.

Row sums of A295027 (shifted).

Main diagonal of A295028 (shifted).

Programs

Maple

a:= n-> add(Stirling1(n+1, k)*(k+1)^(k-1), k=0..n+1)/(n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 21 2016

Mathematica

CoefficientList[Series[-(1+LambertW[-Log[1+x]]/Log[1+x])/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

PARI

{a(n)=n!*polcoeff(sum(i=0,n+1,(i+1)^(i-1)*log(1+x +O(x^(n+2) ))^i/i!), n+1)}

PARI

x='x+O('x^30); Vec(serlaplace(-(1+lambertw(-log(1+x))/log(1+x))/x  )) \\ G. C. Greubel, Feb 19 2018

Formula

a(n) = A033917(n+1)/(n+1).

E.g.f.: A(x) = (1/x)*Sum_{i>=1} (i+1)^(i-1) * log(1+x)^i/i!.

a(n) ~ n^(n-1) / ( exp(n-3/2+exp(-1)/2) * (exp(exp(-1))-1)^(n+1/2) ). - Vaclav Kotesovec, Nov 27 2012