A362835 Expansion of e.g.f. 1/(1 + LambertW(x * log(1-x))).

1, 0, 2, 3, 56, 270, 4704, 43260, 814736, 11356632, 240848640, 4492204200, 108396245088, 2513538490320, 68878522931568, 1896787592514360, 58622475066067200, 1860520458522196800, 64297710768900261888, 2303738717704104464640

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

a(n) = (-1)^n * n! * Sum_{k=0..floor(n/2)} k^k * Stirling1(n-k,k)/(n-k)!.

a(n) ~ c * n^n / (exp(n) * r^n), where r = 0.5123112855238643734867005914814802444318611742227... is the root of the equation r*log(1-r) = -exp(-1) and c = 1/sqrt(1 + exp(1)*r^2/(1-r)) = 0.6371990667116977051617747706059168469728866180606... - Vaclav Kotesovec, Jan 25 2026

Mathematica

Join[{1}, Table[(-1)^n * n!*(Sum[k^k * StirlingS1[n-k, k]/(n-k)!, {k, 1, n/2}]), {n, 1, 20}]]
(* or *)
nmax = 20; CoefficientList[Series[1/(1 + LambertW[x * Log[1-x]]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 23 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(x*log(1-x)))))

Crossrefs

Cf. A318615, A357265, A362836, A362891, A391377.

Sequence in context: A072871 A041709 A361095 * A375688 A371140 A371121

Adjacent sequences: A362832 A362833 A362834 * A362836 A362837 A362838

Keyword

nonn

Author

Seiichi Manyama, May 05 2023

Status

approved