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A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).

%I A345750 #22 Feb 16 2025 08:34:02
%S A345750 1,1,2,9,49,310,2521,25557,290550,3555041,48104901,741103946,
%T A345750 12825399313,240202011881,4747281446090,98808864563065,
%U A345750 2194031697420057,52582450760730398,1357237338948268649
%N A345750 E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k).
%C A345750 Stirling transform of A168243.
%H A345750 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A345750 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A345750 E.g.f.: exp( Sum_{k>=1} A048272(k) * (exp(x) - 1)^k / k ).
%F A345750 a(n) = Sum_{k=0..n} Stirling2(n,k) * A168243(k).
%t A345750 max = 18; 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 A345750 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k))))
%o A345750 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, -sumdiv(k, d, (-1)^d)*(exp(x)-1)^k/k))))
%Y A345750 Cf. A048272, A048993, A168243, A305550, A305987, A345749, A345751.
%K A345750 nonn
%O A345750 0,3
%A A345750 _Seiichi Manyama_, Jun 26 2021