A355338 Expansion of e.g.f.: exp(exp(x) - x^2 - 1).

1, 1, 0, -1, 3, 12, -7, -47, 332, 1347, -2105, -4200, 135457, 474697, -900832, 4682135, 126196787, 439488524, 233313817, 19129265609, 239146712732, 1104038984091, 5891696027079, 89831511761320, 911995655018817, 6253185308181553, 54873149768926624, 653039078246798383

Offset: 0

Author

Vaclav Kotesovec, Jun 29 2022

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..578

Crossrefs

Cf. A000296, A277381, A316778, A355337.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Exp[Exp[x] - x^2 - 1], {x, 0, nmax}], x] * Range[0, nmax]!

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x) - x^2 - 1))) \\ Michel Marcus, Jun 29 2022

Formula

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

a(n) ~ Bell(n) / exp(LambertW(n)^2).

a(0) = a(1) = 1; a(n) = -2 * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jun 29 2022