A177382 - OEIS

%I #11 Oct 02 2020 15:59:49

%S 1,1,4,38,656,17776,695104,37049648,2581673216,227817246976,

%T 24829660693504,3275474443371008,514345822537650176,

%U 94806411271686270976,20269838348763427323904,4975513260049237751994368

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

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

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

%H Vaclav Kotesovec, <a href="/A177382/b177382.txt">Table of n, a(n) for n = 0..150</a>

%F From _Vaclav Kotesovec_, Nov 02 2014: (Start)

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

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

%F (End)

%e E.g.f.: 1 + x + 4*x^2/2! + 38*x^3/3! + 656*x^4/4! + 17776*x^5/5! +...

%e where

%e 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) +...

%t 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 *)

%o (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)}

%Y Cf. A000182, A006752, A335618.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 11 2010