A340473 - OEIS

A340473

a(n) = n! [x^n] W(-W(x))/(-W(x)), where W(x) is the Lambert W function.

1, 1, 1, 7, 13, 321, 31, 42673, -214983, 12251809, -156239909, 6366130761, -135725103227, 5265915854785, -155145910919817, 6318044844152161, -232403136941014799, 10299509100942804033, -446889500139353805773, 21789892230658085847673, -1078684347590588362463619

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OFFSET

0,4

COMMENTS

Let LW(x) = W(-W(x))/(-W(x)) denote the function in the definition and let T(x) = -W(-x) be Euler's tree function A000169, and L(x) = W(-x)/(-x) the labeled tree function A000272, then LW(x) = L(W(x)), and TW(x) := -T(W(-x)) is A097174, and RW(x) := T(-W(-x)) is A207833.

LINKS

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Lambert W-Function.

MAPLE

W := x -> LambertW(x): gf := W(-W(x))/(-W(x)):

ser := series(gf, x, 24): seq(n!*coeff(ser, x, n), n=0..20);

MATHEMATICA

gf := -ProductLog[-ProductLog[x]]/ProductLog[x];

Range[0, 20]! CoefficientList[Series[gf, {x, 0, 20}], x]

PROG

(PARI) my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ Michel Marcus, Jan 09 2021

CROSSREFS