A328124 Let (e*y)^(e*x) = (e*x)^(e*y), y <> x. Numerators of Taylor coefficients of y about x=1.

1, -1, 5, -25, 1243, -1229, 14107, -575927, 4217764, -1408003, 18804662561, -4465808232533, 561757387253483, -55382063966903, 6546034449396991, -52573598131492979, 602340739551273119407, -2476058152523734531, 9618810414948913858931, -139728831996929913343715987, 1341133476946384276848592489

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..300

Mathematics StackExchange, Taylor series about x=e of x^y=y^x

Formula

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.

Example

y = 1 - (x-1) + (5/3)*(x-1)^2 - (25/9)*(x-1)^3 + (1243/270)*(x-1)^4 - (1229/162)*(x-1)^5 + ....

Maple

y:= -x*LambertW(-(1 + ln(x))*exp(-1)/x)/(1 + ln(x)):
S:= series(y, x=1, 31) assuming x>1:
seq(numer(coeff(S, x-1, j)), j=0..30);

Crossrefs

Cf. A328125 (denominators).

Sequence in context: A061583 A278120 A039780 * A033981 A099077 A137113

Adjacent sequences: A328121 A328122 A328123 * A328125 A328126 A328127

Keyword

sign

Author

Robert Israel, Oct 04 2019

Status

approved