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A219541 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} log(1 + k*x).

%I A219541 #15 Mar 27 2019 09:55:53
%S A219541 1,1,3,20,242,4584,124936,4638360,225037200,13820428368,1048006461024,
%T A219541 96171381464256,10503700943629824,1346451508974957696,
%U A219541 200184649396819872768,34167655864475762390784,6635466680845611611326464,1454780635849943337186155520
%N A219541 Expansion of e.g.f.: Sum_{n>=0} Product_{k=1..n} log(1 + k*x).
%H A219541 Vaclav Kotesovec, <a href="/A219541/b219541.txt">Table of n, a(n) for n = 0..144</a>
%F A219541 a(n) ~ exp(1/2) * d^(n+1) * (n!)^2, where d = 1/(Ei(1)-gamma) = 1/(A091725 - A001620) = 0.75878167350772..., where Ei is the second exponential integral and gamma is the Euler-Mascheroni constant. - _Vaclav Kotesovec_, Nov 02 2014
%e A219541 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 20*x^3/3! + 242*x^4/4! + 4584*x^5/5! + ...
%e A219541 where
%e A219541 A(x) = 1 + log(1+x) + log(1+x)*log(1+2*x) + log(1+x)*log(1+2*x)*log(1+3*x) + log(1+x)*log(1+2*x)*log(1+3*x)*log(1+4*x) + ...
%p A219541 a:=series(add(mul(log(1+k*x),k=1..n),n=0..100),x=0,18): seq(n!*coeff(a,x,n),n=0..17); # _Paolo P. Lava_, Mar 27 2019
%t A219541 With[{nmax = 30}, CoefficientList[Series[Sum[Product[Log[1 + j*x], {j, 1, k}], {k,0,3*nmax}], {x,0,nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Sep 04 2018 *)
%o A219541 (PARI) {a(n)=n!*polcoeff(sum(m=0,n,prod(k=1,m,log(1+k*x+x*O(x^n)))),n)}
%o A219541 for(n=0,25,print1(a(n),", "))
%Y A219541 Cf. A091725, A001620.
%K A219541 nonn
%O A219541 0,3
%A A219541 _Paul D. Hanna_, Nov 22 2012