[seqfan] Some Remarkable Series for A193543(n), A193544(n)

[Paul D Hanna]

Dear SeqFans, 
    While investigating series inspired by Ramanujan's Cos/Cosh Identity 
(see: http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html),  
 
I was surprised to find that the following sums are integral for n>0: 
 
A193543(n) = (sqrt(2)*Pi/L) * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi) 
 
A193544(n) = (2*Pi/L) * Sum_{k>=1} (-1)^k*(2*k*Pi/L)^(2*n) / cosh(k*Pi)  
 
where L = Lemniscate constant = 2*(Pi/2)^(3/2) / gamma(3/4)^2 = 2.62205755429... 
 
(see: https://oeis.org/A193543  and  https://oeis.org/A193544). 
 
 
But what then are these sums equal to? 
 
C(n) = (sqrt(2)*Pi/L) * Sum_{k>=1} (2*k*Pi/L)^(2*n-1) / cosh(k*Pi)  
 
D(n) = (2*Pi/L) * Sum_{k>=1} (-1)^k*(2*k*Pi/L)^(2*n-1) / cosh(k*Pi)  
 
where n>0 and L = Lemniscate constant. 
 
Regards, 
  Paul