A244437 E.g.f.: exp( Sum_{n>=1} (1 - exp(-n*x))^n / n ).

1, 1, 4, 41, 845, 30012, 1650475, 130216865, 13944696526, 1945060435587, 342412144747677, 74216506678085290, 19414505134246518741, 6029823819095965829293, 2193174302711080501699684, 923346371767630311443639677, 445468655004100653462280596881, 244137607569262412209821327718964

Offset: 0

Author

Paul D. Hanna, Aug 21 2014

Keywords

• nonn

Comments

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

Examples

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A092552, A243802.

Programs

Mathematica

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 *)

PARI

{a(n) = n!*polcoeff( exp( sum(m=1,n+1, (1 - exp(-m*x +x*O(x^n)))^m / m) ), n)}
for(n=0,20,print1(a(n),", "))

Formula

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.

a(n) ~ (n!)^2 / (2 * sqrt(Pi) * sqrt(1-log(2)) * n^(3/2) * log(2)^(2*n)). - Vaclav Kotesovec, Aug 21 2014