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A346057 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).

%I A346057 #15 Jul 09 2021 08:09:09
%S A346057 1,-1,-1,2,3,14,-55,62,-637,338,-3861,335312,-4499803,43490108,
%T A346057 -246353731,2189950310,-47336985225,1224524919590,-21516426400621,
%U A346057 346681988108648,-4499477383730851,69294602646065900,-1418045089870455795,45246859024830444566
%N A346057 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).
%H A346057 Seiichi Manyama, <a href="/A346057/b346057.txt">Table of n, a(n) for n = 0..451</a>
%F A346057 E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k/k)) ).
%F A346057 E.g.f.: exp( -Sum_{k>=1} A005225(k)*x^k/k! ).
%F A346057 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 A346057 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k)))))
%o A346057 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k)))))
%o A346057 (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 A346057 (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 A346057 Cf. A000587, A005225, A330199, A346055, A346058.
%K A346057 sign
%O A346057 0,4
%A A346057 _Seiichi Manyama_, Jul 02 2021