A012026 - OEIS

%I #19 Jan 29 2018 02:52:50

%S 1,-5,121,-6677,651985,-98741477,21378584137,-6259615830197,

%T 2380487154907681,-1140222272527932869,671453734616884934041,

%U -476714184862146843059285,401522701697360654591942641

%N Expansion of e.g.f. tanh(sin(arctan(x))) = tanh(x/sqrt(1+x^2)) (odd powers only).

%F a(n) = ((2*n+1)!*sum(m=0..n, binomial(n-1/2,n-m)/(2*m+1)!*sum(k=1..2*m+1, (-1)^(n-m+k+1)*k!*2^(2*m+1-k)*Stirling2(2*m+1,k)))). - _Vladimir Kruchinin_, Jun 17 2011

%F E.g.f.: tanh(x/sqrt(1+x^2)) = (x/sqrt(1+x^2))*G(0) where G(k)=  1 - x^2/(x^2 + (1+x^2)*(2*k+1)*(2*k+3)/G(k+1)); (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Aug 06 2012

%F a(n) ~ (2*n-1)! * (-1)^(n+1) * 16 * (4+Pi^2)^(n-3/2) / Pi^(2*n). - _Vaclav Kotesovec_, Feb 02 2015

%e tanh(sin(arctan(x))) = x - (5/3!)*x^3 + (121/5!)*x^5 - (6677/7!)*x^7 + (651985/9!)*x^9 - ...

%t nn = 20; Table[(CoefficientList[Series[Tanh[x/Sqrt[1 + x^2]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* _Vaclav Kotesovec_, Feb 02 2015 *)

%o (Maxima)

%o a(n):=((2*n+1)!*sum(binomial(n-1/2,n-m)/(2*m+1)!*sum((-1)^(n-m+k+1)*k!*2^(2*m+1-k)*stirling2(2*m+1,k),k,1,2*m+1),m,0,n)); /* _Vladimir Kruchinin_, Jun 17 2011 */

%K sign

%O 1,2

%A Patrick Demichel (patrick.demichel(AT)hp.com)