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A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).

%I A347432 #16 Jul 07 2022 07:03:57
%S A347432 1,0,1,4,14,66,397,2626,18797,148238,1281134,11943790,118998365,
%T A347432 1262189748,14203022537,168835162632,2111832477426,27708387132906,
%U A347432 380355066174121,5449577398256414,81316095965242989,1261149374033472626,20293627142875917978,338263983223664609198
%N A347432 E.g.f.: exp( exp(x) * (exp(x) - 1 - x) ).
%C A347432 Exponential transform of A000295.
%F A347432 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A000295(k) * a(n-k).
%F A347432 a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A003725(k) * A143405(n-k).
%F A347432 a(n) ~ n^(n + 1/2) * (exp(exp(r)*(exp(r) - r - 1) - r/2 - n) / (r^(n + 1/2) * sqrt(2*exp(r)*(1 + 2*r) - (2 + r*(4 + r))))), where r = LambertW(n)/2 + (4 + LambertW(n)) * LambertW(n)^(3/2) / (8 * sqrt(n) * (1 + LambertW(n))). - _Vaclav Kotesovec_, Jul 07 2022
%p A347432 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A347432       a(n-j)*binomial(n-1, j-1)*(2^j-j-1), j=1..n))
%p A347432     end:
%p A347432 seq(a(n), n=0..23);  # _Alois P. Heinz_, Sep 02 2021
%t A347432 nmax = 23; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
%t A347432 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
%Y A347432 Cf. A000295, A000296, A003725, A055882, A143405, A347434, A347435.
%K A347432 nonn
%O A347432 0,4
%A A347432 _Ilya Gutkovskiy_, Sep 02 2021