A177382 E.g.f.: Sum_{n>=0} Product_{k=1..n} tan(k*x).

1, 1, 4, 38, 656, 17776, 695104, 37049648, 2581673216, 227817246976, 24829660693504, 3275474443371008, 514345822537650176, 94806411271686270976, 20269838348763427323904, 4975513260049237751994368

Offset: 0

Author

Paul D. Hanna, May 11 2010

Keywords

• nonn

Comments

Compare to an e.g.f. of A000182, the tangent numbers:

Sum_{n>=0} A000182(n)*x^n/n! = Sum_{n>=0} Product_{k=1..n} tanh(k*x).

Examples

			E.g.f.: 1 + x + 4*x^2/2! + 38*x^3/3! + 656*x^4/4! + 17776*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(x)*tan(2*x) + tan(x)*tan(2*x)*tan(3*x) + tan(x)*tan(2*x)*tan(3*x)*tan(4*x) + tan(x)*tan(2*x)*tan(3*x)*tan(4*x)*tan(5*x) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A000182, A006752, A335618.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Sum[Product[Tan[k*x], {k, 1, n}], {n, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 02 2020 *)

PARI

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,tan(k*X)));n!*polcoeff(Egf,n)}

Formula

From Vaclav Kotesovec, Nov 02 2014: (Start)

a(n) ~ (n!)^2 / (sqrt(2) * G^(n+1)).

a(n) ~ Pi * sqrt(2) * n^(2*n+1) / (exp(2*n) * G^(n+1)), where G = A006752 = 0.915965594177219... is Catalan's constant.

(End)