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A305199 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).

%I A305199 #15 Aug 03 2019 12:32:19
%S A305199 1,2,6,28,152,1008,7756,67688,659424,7123776,84154224,1079913888,
%T A305199 14962632384,222447507072,3531920599008,59664827178048,
%U A305199 1067975819206656,20192760528611328,402169396496004864,8414121277765679616,184498963978904644608,4231186653661629843456
%N A305199 Expansion of e.g.f. Product_{k>=1} (1 + x^k/k)/(1 - x^k/k).
%C A305199 Exponential convolution of the sequences A007838 and A007841.
%H A305199 Vaclav Kotesovec, <a href="/A305199/b305199.txt">Table of n, a(n) for n = 0..445</a>
%F A305199 E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (1 + (-1)^(k+1))*x^(j*k)/(k*j^k)).
%F A305199 a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / exp(n + 2*gamma), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Mar 26 2019
%p A305199 a:=series(mul((1+x^k/k)/(1-x^k/k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # _Paolo P. Lava_, Mar 26 2019
%t A305199 nmax = 21; CoefficientList[Series[Product[(1 + x^k/k)/(1 - x^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A305199 nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(1 + (-1)^(k + 1)) x^(j k)/(k j^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%Y A305199 Cf. A007838, A007841, A292358, A292359, A295792.
%K A305199 nonn
%O A305199 0,2
%A A305199 _Ilya Gutkovskiy_, May 27 2018