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

%I #10 Oct 02 2020 15:51:05

%S 1,2,8,50,432,4690,61208,933090,16268640,319249698,6963071784,

%T 167093039122,4374954323216,124108887889522,3791902447022648,

%U 124138462767883202,4335205955612166848,160865445090615444546,6320573384125953811016,262147404448177963790834,11445191965935999115186288

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

%H Vaclav Kotesovec, <a href="/A295759/b295759.txt">Table of n, a(n) for n = 0..265</a>

%F a(n) ~ 2^(n - 1/2) * n! / G^(n+1), where G is the Catalan constant A006752. - _Vaclav Kotesovec_, Oct 02 2020

%e 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 such that

%e 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 nmax = 20;

%t 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 (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 for(n=0,20,print1(a(n),", "))

%Y Cf. A177381, A295758.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 28 2018

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)