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A345751 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k)^(1/k).

%I A345751 #19 Feb 16 2025 08:34:02
%S A345751 1,-1,-2,-3,-3,40,477,4375,45154,486817,5002397,54970652,732601449,
%T A345751 10046371231,113632306694,1051655108629,12585372336141,
%U A345751 202763995934160,-863641466773595,-247388278229558697,-10810815349601723990,-311011007642247422759
%N A345751 E.g.f.: Product_{k>=1} (1 - (exp(x) - 1)^k)^(1/k).
%C A345751 Stirling transform of A028343.
%H A345751 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A345751 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A345751 E.g.f.: exp( -Sum_{k>=1} d(k) * (exp(x) - 1)^k / k ), where d(n) is the number of divisors of n.
%F A345751 a(n) = Sum_{k=0..n} Stirling2(n,k) * A028343(k).
%t A345751 max = 21; Range[0, max]! * CoefficientList[Series[Product[(1 - (Exp[x] - 1)^k)^(1/k), {k, 1, max}], {x, 0, max}], x] (* _Amiram Eldar_, Jun 26 2021 *)
%o A345751 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1-(exp(x)-1)^k)^(1/k))))
%o A345751 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, numdiv(k)*(exp(x)-1)^k/k))))
%Y A345751 Cf. A000005, A028343, A048993, A336100, A345750, A345751, A345752.
%K A345751 sign
%O A345751 0,3
%A A345751 _Seiichi Manyama_, Jun 26 2021