A391377 Expansion of e.g.f. 1/(1 + LambertW(x^3 * log(1-x))).

1, 0, 0, 0, 24, 60, 240, 1260, 88704, 786240, 7171200, 71517600, 2937876480, 51372921600, 807474769920, 12722062953600, 430490106224640, 11120066371814400, 256635966514636800, 5726578592094796800, 191258904944891289600, 6233110401650436864000, 189751022005164912230400

Offset

0,5

Comments

In general, if k >= 1 and e.g.f. = 1/(1 + LambertW(x^k * log(1-x))), then a(n) ~ c * n^n / (exp(n) * r^n), where r is the root of the equation r^k * log(1-r) = -exp(-1) and c = 1/(sqrt(k + exp(1)*r^(k+1)/(1-r))).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Formula

a(n) ~ c * n^n / (exp(n) * r^n), where r = 0.6837029856269721748413229435706375221748341873725... is the root of the equation r^3 * log(1-r) = -exp(-1) and c = 1/(sqrt(3 + exp(1)*r^4/(1-r))) = 0.4527767973401676021460856581728864911399167212456425...

a(n) = n! * Sum_{k=0..floor(n/4)} k^k * |Stirling1(n-3*k,k)|/(n-3*k)!. - Seiichi Manyama, Jan 26 2026

Mathematica

nmax = 25; CoefficientList[Series[1/(1 + LambertW[x^3*Log[1-x]]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 25 2026 *)
Join[{1}, Table[n!* Sum[(k^k)*Abs[StirlingS1[n-3*k, k]]/(n-3*k)!, {k, 1, Floor[n/4]}], {n, 1, 25}]] (* Vincenzo Librandi, Feb 04 2026 *)

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1 + lambertw(x^3 * log(1-x))))) \\ Michel Marcus, Jan 25 2026
(Magma) [Factorial(n)*&+[k^k* Abs(StirlingFirst(n-3*k, k))/Factorial(n-3*k): k in [0..Floor(n/4)] ] : n in [0..24] ]; // Vincenzo Librandi, Feb 04 2026

Crossrefs

Cf. A305981, A362835, A362891.

Sequence in context: A366777 A351504 A356099 * A356971 A376292 A370995

Adjacent sequences: A391374 A391375 A391376 * A391378 A391379 A391380

Keyword

nonn

Author

Vaclav Kotesovec, Jan 25 2026

Status

approved