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A227831 Numerators of coefficients in Taylor series for LambertW(x).

%I A227831 #34 Feb 16 2025 08:33:20
%S A227831 0,1,-1,3,-8,125,-54,16807,-16384,531441,-156250,2357947691,-2985984,
%T A227831 1792160394037,-7909306972,320361328125,-35184372088832,
%U A227831 2862423051509815793,-5083731656658,5480386857784802185939,-32000000000000000,41209797661291758429,-244636361793658185164
%N A227831 Numerators of coefficients in Taylor series for LambertW(x).
%C A227831 The denominators are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is A095996 prefixed by 1.
%D A227831 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
%D A227831 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 34.
%H A227831 Alois P. Heinz, <a href="/A227831/b227831.txt">Table of n, a(n) for n = 0..300</a>
%H A227831 R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, <a href="https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf">On the Lambert W Function</a>
%H A227831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H A227831 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F A227831 Numerators of series reversion of x/(Sum_{n=0..infinity} ((-x)^n)/n!). - _Mats Granvik_, Nov 24 2013
%e A227831 0, 1, -1, 3/2, -8/3, 125/24, -54/5, 16807/720, -16384/315, 531441/4480, -156250/567, 2357947691/3628800, -2985984/1925, ...
%p A227831 series(LambertW(x),x,30); # _N. J. A. Sloane_, Jan 08 2021
%t A227831 Numerator[CoefficientList[Series[LambertW[x], {x, 0, 22}], x]] (* _Mats Granvik_, Nov 24 2013 *)
%t A227831 Numerator[CoefficientList[InverseSeries[Series[x/Sum[((-x)^n)/Factorial[n], {n, 0, 22}], {x, 0, 22}]], x]] (* _Mats Granvik_, Nov 24 2013 *)
%Y A227831 Cf. A095996. See also A036504/A036503.
%K A227831 sign,frac
%O A227831 0,4
%A A227831 _N. J. A. Sloane_, Aug 01 2013