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%I #7 Oct 02 2020 15:52:34
%S 1,1,3,15,113,1105,13219,187103,3058113,56675297,1174295267,
%T 26898243439,674916701169,18409502066097,542373965958595,
%U 17164148092886207,580677914417571585,20913258579319759041,798876414332323236931,32261582928825038942671,1373304514339211081661169
%N O.g.f.: Sum_{n>=0} Product_{k=1..n} tan( (2*k-1)*arctan(x) ).
%H Vaclav Kotesovec, <a href="/A295758/b295758.txt">Table of n, a(n) for n = 0..220</a>
%F a(n) ~ 2^n * n^n / (exp(n) * G^(n + 1/2)), where G is the Catalan constant A006752. - _Vaclav Kotesovec_, Oct 02 2020
%e O.g.f: A(x) = 1 + x + 3*x^2 + 15*x^3 + 113*x^4 + 1105*x^5 + 13219*x^6 + 187103*x^7 + 3058113*x^8 + 56675297*x^9 + 1174295267*x^10 + ...
%e such that
%e A(x) = 1 + x + x*tan(3*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x))*tan(7*arctan(x)) + x*tan(3*arctan(x))*tan(5*arctan(x))*tan(7*arctan(x))*tan(9*arctan(x)) + ...
%o (PARI) {a(n)=local(X=x+x*O(x^n), Gf); Gf=sum(m=0, n, prod(k=1, m, tan((2*k-1)*atan(X)))); polcoeff(Gf, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A177381, A295759.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 28 2018
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