A009198 E.g.f. exp(log(1+x)/exp(x)).

1, 1, -2, 0, 18, -85, 155, 1057, -13636, 86361, -271783, -1518539, 35743378, -376883351, 2663534939, -8383983593, -117620643336, 2907896789457, -39364767602331, 389671721253889, -2504052803057234, -3842812946739679, 465976686530941599

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

For n>=40 the +- signs alternate regularly. - Vaclav Kotesovec, May 13 2022

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..450

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^(Log[1+x]/E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 13 2022 *)

Maxima

a(n):=if n=0 then 1 else sum(sum(binomial(n,i)*(stirling1(i,k)*k^(n-i)*(-1)^(n-i)),i,k,n),k,1,n); /* Vladimir Kruchinin, Jun 29 2011 */

Formula

a(n)=sum(k=1..n, sum(i=k..n, binomial(n,i)*(stirling1(i,k)*k^(n-i)*(-1)^(n-i)))), n > 0, a(0)=1. - Vladimir Kruchinin, Jun 29 2011

a(n) ~ n! * (-1)^n / (Gamma(-exp(1)) * n^(1+exp(1))) * (1 - (exp(2) + exp(1)) * log(n)/n). - Vaclav Kotesovec, May 13 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997