%I #16 Oct 07 2019 03:04:13
%S 1,1,3,9,270,162,1134,28350,127575,26244,216513000,31827411000,
%T 2482538058000,151992126000,11171421261000,55857106305000,
%U 398819739017700000,1022614715430000,2479933649891880000,22505397872768811000000,135032387236612866000000,51557820581252185200000,752545881176354010900000
%N Let (e*y)^(e*x) = (e*x)^(e*y), y <> x. Denominators of Taylor coefficients of y about x=1.
%H Robert Israel, <a href="/A328125/b328125.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(denom(coeff(S, x-1, j)), j=0..30);
%Y Cf. A328124 (numerators).
%K nonn
%O 0,3
%A _Robert Israel_, Oct 04 2019