A305304 - OEIS

%I #11 Mar 27 2019 03:52:42

%S 1,1,2,9,52,405,3786,42301,542984,7924041,129110230,2327399481,

%T 45940938924,986045445853,22856850513602,569163515043285,

%U 15150885843083536,429364157810169105,12905794670246364078,410108007771441394129,13736898888997174964660,483740530150449507164901,17866185834825657429606682

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

%C Inverse Lah transform of A000312.

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

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^k*n!/k!.

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

%p a:=series(1/(1+LambertW(-x/(1+x))),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, Mar 26 2019

%t nmax = 22; CoefficientList[Series[1/(1 + LambertW[-x/(1 + x)]), {x, 0, nmax}], x] Range[0, nmax]!

%t Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] k^k n!/k!, {k, n}], {n, 22}]]

%Y Cf. A000312, A052871, A060356, A305276.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Aug 18 2018