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A264234 Numerators of the coefficients in the expansion of 1/W(x) - 1/x where W(x) is the Lambert W function.

%I A264234 #30 Sep 08 2022 08:46:14
%S A264234 1,-1,2,-9,32,-625,324,-117649,131072,-4782969,1562500,-25937424601,
%T A264234 35831808,-23298085122481,110730297608,-4805419921875,562949953421312,
%U A264234 -48661191875666868481,91507169819844,-104127350297911241532841,640000000000000000,-865405750887126927009
%N A264234 Numerators of the coefficients in the expansion of 1/W(x) - 1/x where W(x) is the Lambert W function.
%C A264234 If prefixed by an additional 1, numerators of coefficients of expansion of exp(W(x)). - _N. J. A. Sloane_, Jan 08 2021
%H A264234 G. C. Greubel, <a href="/A264234/b264234.txt">Table of n, a(n) for n = 0..388</a>
%H A264234 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F A264234 a(n) = (-1)^n*numerator(g(n)) where g(n) = n^n/n!.
%F A264234 a(n) = (-1)^n*denominator(h(n)) where h(n) = Sum_{k=0..n-1}(n!*n^k)/(k!*n^n).
%e A264234 Coefficients of expansion of exp(W(x)) are 1, 1, -1/2, 2/3, -9/8, 32/15, -625/144, 324/35, -117649/5760, 131072/2835, -4782969/44800, ... - _N. J. A. Sloane_, Jan 08 2021
%p A264234 seq(numer((-1)^n*n^n/n!), n = 0..21);
%t A264234 CoefficientList[Series[1/ProductLog[x] - 1/x, {x, 0, 21}], x] // Numerator
%o A264234 (PARI) vector(22, n, n--; (-1)^n*numerator(n^n/n!)) \\ _Altug Alkan_, Nov 09 2015
%o A264234 (Magma) [(-1)^n * Numerator(n^n/Factorial(n)): n in [0..50]]; // _G. C. Greubel_, Nov 14 2017
%Y A264234 Denominators in A264235.
%Y A264234 Cf. A036505.
%K A264234 sign,frac
%O A264234 0,3
%A A264234 _Peter Luschny_, Nov 09 2015