A305819 - OEIS

%I #20 Feb 05 2022 01:48:51

%S 1,1,3,17,132,1334,16442,239994,4041776,77183328,1647541632,

%T 38877352392,1004869488048,28234217634024,856830099396840,

%U 27930093941047464,973269467390922240,36104568839480990400,1420556927968241858880,59088101641333114906944,2590680379402887359111424

%N Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

%C Inverse Stirling transform of A000312.

%H Seiichi Manyama, <a href="/A305819/b305819.txt">Table of n, a(n) for n = 0..397</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*k^k.

%F a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - _Vaclav Kotesovec_, Aug 18 2018

%p a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # _Paolo P. Lava_, Mar 26 2019

%t nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]

%o (PARI) a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ _Seiichi Manyama_, Feb 05 2022

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ _Seiichi Manyama_, Feb 05 2022

%Y Cf. A000312, A052807, A277489, A282190, A305981.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 18 2018