A308517 Expansion of e.g.f. exp(1 - exp(x/(1 - x))).

1, -1, -2, -5, -11, 18, 711, 10113, 125042, 1485627, 17151083, 185932580, 1665928529, 4570649471, -349942007986, -14532197609433, -433111168649251, -11579368513540914, -293948221716443209, -7208510256850719447, -170577027262193604678, -3823168355141657356481, -76959686241473750407701

Offset: 0

Author

Ilya Gutkovskiy, Jun 03 2019

Keywords

• sign

Comments

Lah transform of A000587 (complementary Bell numbers).

Crossrefs

Cf. A000587, A084357.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Exp[1 - Exp[x/(1 - x)]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, -1] n!/k!, {k, 0, n}]; Table[a[n], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A000587(k)*n!/k!.