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%I #18 Jul 07 2021 11:24:23
%S 1,-1,-1,3,1,17,-119,165,1191,-21989,169527,-317837,-7182779,54452161,
%T 292654649,-4320853051,-46883217705,728176373539,9943868087879,
%U -166076498591597,-2748733072385043,65290726021558089,151614363753006601,-11661992771499644571
%N Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k))^(1/k!).
%H Seiichi Manyama, <a href="/A346039/b346039.txt">Table of n, a(n) for n = 0..464</a>
%F E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k))/k! ).
%F E.g.f.: exp( -Sum_{k>=1} A121860(k)*x^k/k! ).
%F a(n) = -(n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)!)) * a(n-k)/(n-k)! for n > 0.
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k))^(1/k!))))
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (1-exp(x^k))/k!))))
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)!))*x^k))))
%o (PARI) a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)!))*a(n-k)/(n-k)!));
%Y Cf. A000587, A121860, A330199, A345762, A346037, A346058.
%K sign
%O 0,4
%A _Seiichi Manyama_, Jul 02 2021
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