A360175 a(n) = Sum_{k=0..n} (-1)^(n-k)*(n!/k!) * [x^n] (1 - exp(-LambertW(x*exp(-x))))^k.

1, 1, 6, 53, 647, 10092, 191915, 4309769, 111682044, 3281731611, 107860953795, 3921762633846, 156322429050397, 6779458454252941, 317841794915501862, 16020304439710056785, 863955306007083830051, 49641711131738762890764, 3027776406780183894833791, 195382900651186641677702197

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Formula

a(n) = Sum_{k=0..n} |A360176(n, k)|.

a(n) ~ c * n^(n-1) / (exp(n) * LambertW(exp(-1))^n), where c = 17.13480404664326452689180574722095702380118... - Vaclav Kotesovec, Jan 31 2026

Maple

egf := k -> (1 - exp(-LambertW(x*exp(-x))))^k / k!:
ser := k -> series(egf(k), x, 22):
T := (n, k) -> (-1)^(n-k)*n!*coeff(ser(k), x, n):
seq(add(T(n, k), k = 0..n), n = 0..19);

Mathematica

nmax = 20; Table[n! * Sum[(-1)^(n-k)/k! * SeriesCoefficient[(1 - E^(-LambertW[x*E^(-x)]))^k, {x, 0, n}], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Jan 30 2026 *)

Crossrefs

Cf. A360176.

Sequence in context: A109092 A068416 A360231 * A221413 A145003 A386927

Adjacent sequences: A360172 A360173 A360174 * A360176 A360177 A360178

Keyword

nonn

Author

Peter Luschny, Jan 29 2023

Status

approved