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

1, 2, 8, 50, 432, 4690, 61208, 933090, 16268640, 319249698, 6963071784, 167093039122, 4374954323216, 124108887889522, 3791902447022648, 124138462767883202, 4335205955612166848, 160865445090615444546, 6320573384125953811016, 262147404448177963790834, 11445191965935999115186288

Offset: 0

Author

Paul D. Hanna, Jan 28 2018

Keywords

• nonn

Examples

			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 + + ...
such that
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)) + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..265

Crossrefs

Cf. A177381, A295758.

Programs

Mathematica

nmax = 20;
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 *)

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)}
for(n=0,20,print1(a(n),", "))

Formula

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