A308519 Expansion of e.g.f. exp(exp(1 - exp(x)) - 1).

1, -1, 1, 0, -2, 2, 9, -24, -80, 339, 1127, -6438, -22051, 179670, 705969, -6316587, -34059363, 234694313, 1788358694, -10539103220, -107105535812, 679821765685, 9404735956848, -39383677719404, -965379929608361, 375758547940149, 86199829280567638, 158307964563289211

Offset: 0

Author

Ilya Gutkovskiy, Jun 03 2019

Keywords

• sign

Comments

Exponential transform of A000587 (complementary Bell numbers).

Crossrefs

Cf. A000110, A000258, A000587, A308518.

Programs

Mathematica

nmax = 27; CoefficientList[Series[Exp[Exp[1 - Exp[x]] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n, k] BellB[k], {k, 0, n}], {n, 0, 27}]
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, -1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 27}]

Formula

a(n) = Sum_{k=0..n} (-1)^k*Stirling2(n,k)*A000110(k).

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