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

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

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

Formula

a(n) ~ -(-1)^n * LambertW(1)^2 * n^(n-1) / (1 + LambertW(1)). - Vaclav Kotesovec, Jan 29 2026

a(0) = 1; a(n) = Sum_{k=0..n-1} (-n)^(n-1-k) * (k+2)^k * binomial(n-1,k). - Seiichi Manyama, Jun 01 2026

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

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

Cf. A000169, A000272.

Sequence in context: A139188 A064119 A224984 * A228030 A082706 A342931

Adjacent sequences: A340470 A340471 A340472 * A340474 A340475 A340476

Keyword

sign

Author

Peter Luschny, Jan 08 2021

Status

approved