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%I #8 Jul 03 2021 01:37:13
%S 1,-1,-2,1,1,98,-39,3225,1226,6459,12473,821830,-214739887,-201448561,
%T -8997850614,-514986723363,-1310942141971,-26465356716946,
%U -931753364233567,-1858534483400559,167210272584038942,-7112146717031426801,312288595642509829797
%N Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k)).
%H Seiichi Manyama, <a href="/A330199/b330199.txt">Table of n, a(n) for n = 0..450</a>
%F E.g.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = e.g.f. of complementary Bell numbers (A000587).
%F E.g.f.: exp(-Sum_{j>=1} Sum_{i>=1} x^(i*j) / i!).
%F a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * A057625(k) * a(n-k).
%t nmax = 22; CoefficientList[Series[Product[Exp[1 - Exp[x^k]], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] k! DivisorSum[k, 1/#! &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
%Y Cf. A000587, A057625, A209903.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 05 2019
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