A328125 Let (e*y)^(e*x) = (e*x)^(e*y), y <> x. Denominators of Taylor coefficients of y about x=1.

1, 1, 3, 9, 270, 162, 1134, 28350, 127575, 26244, 216513000, 31827411000, 2482538058000, 151992126000, 11171421261000, 55857106305000, 398819739017700000, 1022614715430000, 2479933649891880000, 22505397872768811000000, 135032387236612866000000, 51557820581252185200000, 752545881176354010900000

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(denom(coeff(S, x-1, j)), j=0..30);

Crossrefs

Cf. A328124 (numerators).

Sequence in context: A211898 A318970 A132516 * A128450 A361033 A132562

Adjacent sequences: A328122 A328123 A328124 * A328126 A328127 A328128

Keyword

nonn

Author

Robert Israel, Oct 04 2019

Status

approved