A392897 Expansion of e.g.f. 1/(1 + LambertW(-x^3 * (exp(x) - 1))).

1, 0, 0, 0, 24, 60, 120, 210, 80976, 726264, 4234320, 19959390, 2238003240, 42343743156, 491476267464, 4417076666730, 256926209383200, 7818120233767920, 149407089980283936, 2171024442233343414, 89679067883324589240, 3606536452635375767340, 100275124468994359989720

Offset

0,5

Comments

In general, if k>=1 and e.g.f. = 1/(1 + LambertW(-x^k * (exp(x) - 1))), then a(n) ~ n^n / (sqrt((k+r)*exp(r) - k) * r^(n + k/2) * exp(n + 1/2)), where r is the root of the equation exp(1+r) - exp(1) = 1/r^k. - Vaclav Kotesovec, Jan 26 2026

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} k^k * Stirling2(n-3*k,k)/(n-3*k)!.

a(n) ~ n^n / (sqrt((3+r)*exp(r) - 3) * r^(n + 3/2) * exp(n + 1/2)), where r = 0.7090388719608318425589554360471409259927508877244... is the root of the equation exp(1+r) - exp(1) = 1/r^3. - Vaclav Kotesovec, Jan 26 2026

Mathematica

Join[{1}, Table[n!*Sum[ k^k* Abs[StirlingS2[n-3*k, k]/(n-3*k)!], {k, 1, Floor[n/4]}], {n, 1, 23}]] (* Vincenzo Librandi, Jan 26 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^3*(exp(x)-1)))))
(Magma) [Factorial(n)* &+[k^k *Abs(StirlingSecond(n-3*k, k))/Factorial(n-3*k) : k in [0..Floor(n/4)] ] : n in [0..23] ]; // Vincenzo Librandi, Jan 26 2026

Crossrefs

Cf. A282190, A362836, A362892.

Sequence in context: A366751 A358014 A356950 * A356892 A376293 A370990

Adjacent sequences: A392894 A392895 A392896 * A392898 A392899 A392900

Keyword

nonn

Author

Seiichi Manyama, Jan 26 2026

Status

approved