A331796 E.g.f.: (exp(x) - 1) * exp(1 - exp(x)) / (2 - exp(x)).

0, 1, 1, 4, 27, 201, 1730, 17403, 200753, 2607034, 37614509, 596935373, 10334325760, 193820393781, 3914731176005, 84716449797164, 1955520065429447, 47960724916860501, 1245468600257306394, 34139796085144434199, 985066290121984334613

Offset: 0

Author

Ilya Gutkovskiy, Jan 26 2020

Keywords

• nonn

Comments

Stirling transform of A000240.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424

Crossrefs

Cf. A000240, A000587, A000670, A064898, A331797.

Programs

Maple

g:= proc(n) option remember;
     `if`(n=0, 0, n*(g(n-1)-(-1)^n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 23 2023

Mathematica

nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[1 - Exp[x]]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
A000240[n_] := n! Sum[(-1)^k/k!, {k, 0, n - 1}]; a[n_] := Sum[StirlingS2[n, k] A000240[k], {k, 0, n}]; Table[a[n], {n, 0, 20}]
Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k, -1], {k, 1, n}], {n, 0, 20}]

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k) * A000240(k).

a(n) = Sum_{k=1..n} binomial(n,k) * A000670(k) * A000587(n-k).

a(n) ~ n! * exp(-1) / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Jan 26 2020