A331796 - OEIS

%I #14 Jun 23 2023 18:12:32

%S 0,1,1,4,27,201,1730,17403,200753,2607034,37614509,596935373,

%T 10334325760,193820393781,3914731176005,84716449797164,

%U 1955520065429447,47960724916860501,1245468600257306394,34139796085144434199,985066290121984334613

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

%C Stirling transform of A000240.

%H Alois P. Heinz, <a href="/A331796/b331796.txt">Table of n, a(n) for n = 0..424</a>

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

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

%F a(n) ~ n! * exp(-1) / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Jan 26 2020

%p g:= proc(n) option remember;

%p      `if`(n=0, 0, n*(g(n-1)-(-1)^n))

%p     end:

%p b:= proc(n, m) option remember; `if`(n=0,

%p       g(m), m*b(n-1, m)+b(n-1, m+1))

%p     end:

%p a:= n-> b(n, 0):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 23 2023

%t nmax = 20; CoefficientList[Series[(Exp[x] - 1) Exp[1 - Exp[x]]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%t Table[(1/2) Sum[Binomial[n, k] HurwitzLerchPhi[1/2, -k, 0] BellB[n - k, -1], {k, 1, n}], {n, 0, 20}]

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

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Jan 26 2020