A277458 Expansion of e.g.f. -1/(1-LambertW(-x)).
-1, 1, 0, 3, 16, 165, 2016, 30415, 539904, 11049129, 256038400, 6627314331, 189517916160, 5933803272397, 201893195083776, 7417376809230375, 292648536838045696, 12341039738944113105, 553942486234823786496, 26369048375194607316019, 1326864458454400696320000
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..385
- Jason Saied, Jeffrey Marshall, Namit Anand, and Eleanor G. Rieffel, General protocols for the efficient distillation of indistinguishable photons, arxiv:2404.14217 [quant-ph], Apr 22 2024. See p. 14.
Programs
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Maple
seq(n!*add((-1)^(k+1)*k*n^(n-k-1)/(n-k)!, k = 1..n), n = 1..20); # Peter Bala, Jul 23 2021
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Mathematica
CoefficientList[Series[-1/(1-LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]! -
PARI
my(x='x+O('x^50)); Vec(serlaplace(-1/(1 - lambertw(-x)))) \\ G. C. Greubel, Nov 07 2017
Formula
a(n) ~ n^(n-1) / 4.
a(n) = n!*Sum_{k = 1..n} (-1)^(k+1)*k*n^(n-k-1)/(n-k)! for n >= 1. Cf. A133297. - Peter Bala, Jul 23 2021
a(n) = (-1)^(n+1)*U(1-n, -n, -n) where U is the Kummer U function. - Peter Luschny, Jan 23 2025