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A346058 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k!)).

%I A346058 #16 Jul 08 2021 10:59:52
%S A346058 1,-1,-1,3,4,2,-69,-185,596,1482,22051,-8341,-450570,-1503596,
%T A346058 -23829233,144974757,150086353,4859956733,51013196234,-504522222442,
%U A346058 2572161050316,-58533039862692,69278113622988,342581575176372,-25348876024693055,661312712021911319
%N A346058 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k!)).
%H A346058 Seiichi Manyama, <a href="/A346058/b346058.txt">Table of n, a(n) for n = 0..537</a>
%F A346058 E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k/k!)) ).
%F A346058 E.g.f.: exp( -Sum_{k>=1} A038041(k)*x^k/k! ).
%F A346058 a(n) = -(n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!^d)) * a(n-k)/(n-k)! for n > 0.
%o A346058 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k!)))))
%o A346058 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k!)))))
%o A346058 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!^d))*x^k))))
%o A346058 (PARI) a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!^d))*a(n-k)/(n-k)!));
%Y A346058 Cf. A000587, A038041, A330199, A346056, A346057.
%K A346058 sign
%O A346058 0,4
%A A346058 _Seiichi Manyama_, Jul 02 2021