A059606 Expansion of (1/2)*(exp(2*x)-1)*exp(exp(x)-1).

0, 1, 4, 16, 68, 311, 1530, 8065, 45344, 270724, 1709526, 11376135, 79520644, 582207393, 4453142140, 35500884556, 294365897104, 2533900264547, 22604669612078, 208656457858161, 1990060882027600

Offset: 0

Author

Vladeta Jovovic, Jan 29 2001

Keywords

• nonn
• easy

Comments

Starting (1, 4, 16, 68, 311, ...), = A008277 * A000217, i.e., the product of the Stirling2 triangle and triangular series. - Gary W. Adamson, Jan 31 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Vladeta Jovovic, More information.

Crossrefs

Cf. A000110, A005493, A059604, A059605.

Cf. A035098.

Cf. A008277, A000217.

Programs

Maple

s := series(1/2*(exp(2*x)-1)*exp(exp(x)-1), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

Mathematica

With[{nn=20},CoefficientList[Series[((Exp[2x]-1)Exp[Exp[x]-1])/2,{x,0,nn}] ,x] Range[0,nn]!] (* Harvey P. Dale, Nov 10 2011 *)

Formula

a(n) = Sum_{i=0..n} Stirling2(n, i)*binomial(i+1, 2).

a(n) = (1/2)*(Bell(n+2)-Bell(n+1)-Bell(n)). - Vladeta Jovovic, Sep 23 2003

G.f.: Sum_{k>=1} (k*(k + 1)/2)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018

a(n) ~ n^2 * Bell(n) / (2*LambertW(n)^2) * (1 - LambertW(n)/n). - Vaclav Kotesovec, Jul 28 2021