A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470

Offset: 0

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

• nonn

Crossrefs

Cf. A000670, A001861, A052555, A080253, A083355, A216794, A346417, A346432.

Programs

Mathematica

nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001861(k) * a(n-k).

a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * A000670(k).

a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021