A277489 Expansion of e.g.f. -LambertW(-log(1+x)).

0, 1, 1, 5, 26, 224, 2244, 28496, 417976, 7122384, 136770960, 2937770472, 69626588976, 1806936836184, 50936933449752, 1550292926398680, 50661309325357824, 1769286989373994752, 65762170385201959680, 2591979585702305271552, 107982615297265761991680

Offset: 0

Author

Vaclav Kotesovec, Oct 17 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..395
• Index entries for reversions of series

Crossrefs

Cf. A001865, A052807, A133297.

Cf. A357337, A357338.

Programs

Mathematica

CoefficientList[Series[-LambertW[-Log[1+x]], {x, 0, 20}], x] * Range[0, 20]!
Table[Sum[StirlingS1[n, k]*k^(k-1), {k, 1, n}], {n, 0, 20}]

PARI

my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-log(1+x))))) \\ G. C. Greubel, Jun 21 2017

Formula

a(n) = Sum_{k=1..n} Stirling1(n, k)*k^(k-1).

a(n) ~ (exp(exp(-1))-1)^(1/2-n) * exp(-exp(-1)/2+1/2-n) * n^(n-1).

E.g.f.: Series_Reversion( exp(x * exp(-x)) - 1 ). - Seiichi Manyama, Sep 10 2024