login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = (-1)^n*n!*[x^n] exp(-x)/(1 + log(1+x)).
12

%I #27 Aug 30 2024 21:28:03

%S 1,2,6,27,167,1310,12394,137053,1733325,24670114,390204086,6789564639,

%T 128884276179,2650516064222,58701784670138,1392959655437473,

%U 35257885037803417,948208649740610466,27000743345935785670,811575543670852269347,25677856392014665436799

%N a(n) = (-1)^n*n!*[x^n] exp(-x)/(1 + log(1+x)).

%C Row sums of A291978.

%H Seiichi Manyama, <a href="/A291979/b291979.txt">Table of n, a(n) for n = 0..418</a>

%F a(n) ~ sqrt(2*Pi) * n^(n+1/2) * exp(1 - exp(-1)) / (exp(1)-1)^(n+1). - _Vaclav Kotesovec_, Sep 18 2017

%F a(n) = 1 + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - _Ilya Gutkovskiy_, Apr 26 2021

%F a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k)*j!*A132393(k,j). - _Fabian Pereyra_, Aug 29 2024

%p a_list := proc(n) exp(-x)/(1 + log(1+x)): series(%, x, n+1):

%p seq((-1)^k*k!*coeff(%, x, k), k=0..n) end: a_list(20);

%t nmax = 20; CoefficientList[Series[E^(-x)/(1 + Log[1+x]), {x, 0, nmax}], x] * Range[0, nmax]! * (-1)^Range[0, nmax] (* _Vaclav Kotesovec_, Sep 18 2017 *)

%o (PARI) N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+log(1-x)))) \\ _Seiichi Manyama_, Oct 20 2021

%Y Cf. A132393, A291978, A343707, A343709, A343710.

%K nonn

%O 0,2

%A _Peter Luschny_, Sep 16 2017