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A177385 E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).

%I A177385 #19 Nov 06 2014 11:43:17
%S A177385 1,1,4,37,616,16081,605164,31011457,2076192976,175951716481,
%T A177385 18411425885524,2331339303739777,351341718484191736,
%U A177385 62144180030978834881,12748469150999320273084,3002313213700366145858497
%N A177385 E.g.f.: Sum_{n>=0} Product_{k=1..n} sinh(k*x).
%C A177385 Compare to the e.g.f. for A002105, the reduced tangent numbers:
%C A177385 . Sum_{n>=0} A002105(n+1)*x^n/n! = Sum_{n>=0} Product_{k=1..n} tanh(k*x).
%C A177385 Limit n->infinity n!*A177386(n) / (2^n*A177385(n)) = 1. - _Vaclav Kotesovec_, Nov 06 2014
%H A177385 Vaclav Kotesovec, <a href="/A177385/b177385.txt">Table of n, a(n) for n = 0..250</a>
%F A177385 a(n) ~ c * d^n * (n!)^2, where d = A249748 = 1.04689919262595424111342518325311817976789046475647184115584744582777576864..., c = 0.880333778211172907563073031129920597506533414605109200048966773434616066... . - _Vaclav Kotesovec_, Nov 04 2014
%e A177385 E.g.f: A(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 616*x^4/4! +...
%e A177385 A(x) = 1 + sinh(x) + sinh(x)*sinh(2x) + sinh(x)*sinh(2x)*sinh(3x) + ...
%t A177385 Table[n!*SeriesCoefficient[Sum[Product[Sinh[k*x],{k,1,j}],{j,0,n}],{x,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 01 2014 *)
%t A177385 nn=20; tab = ConstantArray[0,nn]; tab[[1]] = Series[Sinh[x],{x,0,nn}]; Do[tab[[k]] = Series[tab[[k-1]]*Sinh[k*x],{x,0,nn}],{k,2,nn}]; Flatten[{1,Rest[CoefficientList[Sum[tab[[k]],{k,1,nn}],x] * Range[0,nn]!]}] (* _Vaclav Kotesovec_, Nov 04 2014 (more efficient) *)
%o A177385 (PARI) {a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,sinh(k*X)));n!*polcoeff(Egf,n)}
%Y A177385 Cf. A249698, A177386, A249564, A249748.
%K A177385 nonn
%O A177385 0,3
%A A177385 _Paul D. Hanna_, May 15 2010