A340473 - OEIS

%I #22 Jan 09 2021 07:52:23

%S 1,1,1,7,13,321,31,42673,-214983,12251809,-156239909,6366130761,

%T -135725103227,5265915854785,-155145910919817,6318044844152161,

%U -232403136941014799,10299509100942804033,-446889500139353805773,21789892230658085847673,-1078684347590588362463619

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

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

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

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

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

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

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(lambertw(-lambertw(x))/(-lambertw(x)))) \\ _Michel Marcus_, Jan 09 2021

%Y Cf. A097174, A177885, A207833, A227176, A340474.

%Y Cf. A000169, A000272.

%K sign

%O 0,4

%A _Peter Luschny_, Jan 08 2021