%I #32 Mar 31 2016 11:58:32

%S 1,1,4,41,845,30012,1650475,130216865,13944696526,1945060435587,

%T 342412144747677,74216506678085290,19414505134246518741,

%U 6029823819095965829293,2193174302711080501699684,923346371767630311443639677,445468655004100653462280596881,244137607569262412209821327718964

%N E.g.f.: exp( Sum_{n>=1} (1 - exp(-n*x))^n / n ).

%C Compare to: exp( Sum_{n>=1} (1 - exp(-x))^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers (A000670).

%H Vaclav Kotesovec, <a href="/A244437/b244437.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: exp( Sum_{n>=1} A092552(n)*x^n/n! ), where A092552(n) = Sum_{k=1..n} k!*(k-1)! * Stirling2(n, k)^2.

%F a(n) ~ (n!)^2 / (2 * sqrt(Pi) * sqrt(1-log(2)) * n^(3/2) * log(2)^(2*n)). - _Vaclav Kotesovec_, Aug 21 2014

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 41*x^3/3! + 845*x^4/4! + 30012*x^5/5! +...

%e where

%e log(A(x)) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 + (1-exp(-4*x))^4/4 + (1-exp(-5*x))^5/5 + (1-exp(-6*x))^6/6 +...

%e Explicitly,

%e log(A(x)) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! + 1441923*x^6/6! +...+ A092552(n)*x^n/n! +...

%t max = 20; s = Exp[Sum[(1 - Exp[-n x])^n/n, {n, 1, max}]] + O[x]^max; CoefficientList[s, x] Range[0, max-1]! (* _Jean-François Alcover_, Mar 31 2016 *)

%o (PARI) {a(n) = n!*polcoeff( exp( sum(m=1,n+1, (1 - exp(-m*x +x*O(x^n)))^m / m) ), n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A092552, A243802.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 21 2014