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A295759 O.g.f.: Sum_{n>=0} Product_{k=1..n} tan( (2*k)*arctan(x) ).

%I A295759 #10 Oct 02 2020 15:51:05
%S A295759 1,2,8,50,432,4690,61208,933090,16268640,319249698,6963071784,
%T A295759 167093039122,4374954323216,124108887889522,3791902447022648,
%U A295759 124138462767883202,4335205955612166848,160865445090615444546,6320573384125953811016,262147404448177963790834,11445191965935999115186288
%N A295759 O.g.f.: Sum_{n>=0} Product_{k=1..n} tan( (2*k)*arctan(x) ).
%H A295759 Vaclav Kotesovec, <a href="/A295759/b295759.txt">Table of n, a(n) for n = 0..265</a>
%F A295759 a(n) ~ 2^(n - 1/2) * n! / G^(n+1), where G is the Catalan constant A006752. - _Vaclav Kotesovec_, Oct 02 2020
%e A295759 O.g.f: A(x) = 1 + 2*x + 8*x^2 + 50*x^3 + 432*x^4 + 4690*x^5 + 61208*x^6 + 933090*x^7 + 16268640*x^8 + 319249698*x^9 + 6963071784*x^10 + + ...
%e A295759 such that
%e A295759 A(x) = 1 + tan(2*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x))*tan(8*arctan(x)) + tan(2*arctan(x))*tan(4*arctan(x))*tan(6*arctan(x))*tan(8*arctan(x))*tan(10*arctan(x)) + ...
%t A295759 nmax = 20;
%t A295759 Sum[Product[Tan[2 k ArcTan[x]], {k, 1, n}] , {n, 0, nmax}] + O[x]^(nmax+1) // CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 02 2020 *)
%o A295759 (PARI) {a(n)=local(X=x+x*O(x^n), Gf); Gf=sum(m=0, n, prod(k=1, m, tan((2*k)*atan(X)))); polcoeff(Gf, n)}
%o A295759 for(n=0,20,print1(a(n),", "))
%Y A295759 Cf. A177381, A295758.
%K A295759 nonn
%O A295759 0,2
%A A295759 _Paul D. Hanna_, Jan 28 2018