The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A337734

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

#9 by N. J. A. Sloane at Mon Sep 28 02:03:34 EDT 2020

STATUS

proposed

approved

#8 by Wesley Ivan Hurt at Thu Sep 17 15:04:58 EDT 2020

STATUS

editing

proposed

#7 by Wesley Ivan Hurt at Thu Sep 17 15:04:45 EDT 2020

CROSSREFS

Cf. A337735 (denominators).

STATUS

proposed

editing

#6 by Max Alekseyev at Thu Sep 17 14:47:46 EDT 2020

STATUS

editing

proposed

#5 by Max Alekseyev at Thu Sep 17 14:47:42 EDT 2020

NAME

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

COMMENTS

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

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.

STATUS

proposed

editing

#4 by Max Alekseyev at Thu Sep 17 14:43:48 EDT 2020

STATUS

editing

proposed

#3 by Max Alekseyev at Thu Sep 17 14:43:39 EDT 2020

NAME

allocated for Max AlekseyevNumerators of the coefficient in the expansion of li^{-1}(x)/x in powers of 1/LambertW(-1,-e/x).

DATA

-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) * LambertW(-1,-e/x)^(-n).

LINKS

Martin et al., <a href="https://mathoverflow.net/q/278626">Expansion of inverse logarithmic integral in terms of lambert w</a>, MathOverflow, 2017.

FORMULA

Function f(t) := Sum_{n>=1} a(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)

KEYWORD

allocated

frac,sign

AUTHOR

Max Alekseyev, Sep 17 2020

STATUS

approved

editing

#2 by Max Alekseyev at Thu Sep 17 14:08:59 EDT 2020

KEYWORD

allocating

allocated

#1 by Max Alekseyev at Thu Sep 17 14:08:59 EDT 2020

NAME

allocated for Max Alekseyev

KEYWORD

allocating

STATUS

approved