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%I #16 Oct 07 2019 03:04:25
%S 1,-1,5,-25,1243,-1229,14107,-575927,4217764,-1408003,18804662561,
%T -4465808232533,561757387253483,-55382063966903,6546034449396991,
%U -52573598131492979,602340739551273119407,-2476058152523734531,9618810414948913858931,-139728831996929913343715987,1341133476946384276848592489
%N Let (e*y)^(e*x) = (e*x)^(e*y), y <> x. Numerators of Taylor coefficients of y about x=1.
%H Robert Israel, <a href="/A328124/b328124.txt">Table of n, a(n) for n = 0..300</a>
%H Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/3373846/taylor-series-about-x-e-of-xy-yx/3373868#3373868">Taylor series about x=e of x^y=y^x</a>
%F y = - (x/log(e*x)) * W(-log(e*x)/(e*x)) where W is the main branch of the Lambert W function for x > 1 and the "-1" branch for x < 1.
%e y = 1 - (x-1) + (5/3)*(x-1)^2 - (25/9)*(x-1)^3 + (1243/270)*(x-1)^4 - (1229/162)*(x-1)^5 + ....
%p y:= -x*LambertW(-(1 + ln(x))*exp(-1)/x)/(1 + ln(x)):
%p S:= series(y, x=1, 31) assuming x>1:
%p seq(numer(coeff(S,x-1,j)),j=0..30);
%Y Cf. A328125 (denominators).
%K sign
%O 0,3
%A _Robert Israel_, Oct 04 2019
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