A337734 Numerators of the coefficients in the expansion of li^{-1}(x)/x in powers of 1/LambertW(-1,-e/x).

-1, 0, 1, -3, 11, -105, 613, -12635, 99677, -1774391, 17582819, -1919343719, 22882040099, -295793507053, 1373607474819, -323119030735871, 20600974525589671, -698062672818463097, 12527062232269129201, -474730436062281169829, 9471193365463611988187, -396898731474190849635703

Offset

-1,4

Comments

li^{-1}(x) / x = Sum_{n>=-1} a(n)/A337735(n) * LambertW(-1,-e/x)^(-n).

Links

Table of n, a(n) for n=-1..20.

Martin et al., Expansion of inverse logarithmic integral in terms of lambert w, MathOverflow, 2017.

Formula

Function f(t) := Sum_{n>=1} a(n)/A337735(n) * t^{n-1} satisfies the differential equation: t^3*f'(t) + t*(1+2*t)*f(t) - (1+t)*log(1-t^2*f(t)) - t = 0 with f(0) = 1.

Maple

Order:=20: dsolve( { t^3*diff(f(t), t) + t*(1+2*t)*f(t) - (1+t)*log(1-t^2*f(t)) - t = 0, f(0)=1 }, f(t), series);

Crossrefs

Cf. A337735 (denominators).

Sequence in context: A121045 A092245 A238446 * A302927 A105413 A241587

Adjacent sequences: A337731 A337732 A337733 * A337735 A337736 A337737

Keyword

frac,sign

Author

Max Alekseyev, Sep 17 2020

Status

approved