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

[Paul D Hanna]

SeqFans,     Consider a slight variation in the series for A193543(n) mentioned in prior email 
by defining  
 a(n) = (sqrt(2)*Pi/L) * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi/2) for n>0 with a(0)=1. 
Then 
a(1) = 9.656854249492380195206754896838792314278687501507792...
a(2) = 279.7645019878171246849621175241310155426885000361870...
a(3) = 19567.04414312283297731727246173743311907357200260546...
a(4) = 2551384.434701383509036123601667024594225613408340447...
a(5) = 534405622.3432607796041264218343605722780425756872021...
a(6) = 164165809659.5750828344220598502172879799939350603429...
a(7) = 69532826885135.18156333436878613710259344193824133284...
a(8) = 38836197110487315.84344087627800470932336285596433235...
a(9) = 27656273940212637887.65845519441121088691142638967441...
a(10) = 24457541968339546203692.3434925846139712563531814594...... 
which I find to begin:   
a(1) = 4 + 4*sqrt(2)) 
a(2) = 144 + 96*sqrt(2) 
a(3) = 9792 + 6912*sqrt(2) 
... 
>From this I conjecture that 
a(n) = b(n) + c(n)*sqrt(2) 
 
Without Plouffe's converter, I can't go much further. 
(BTW, it was by using Plouffe's Inverter that I was able to find  
the constants involved in the sums for A193543(n), A193544(n).)
 
Can someone find more terms of b(n) and c(n)? 
 
Here is my PARI code:\p80 
{a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0,1, 
sqrt(2)*Pi/L*suminf(k=1, (2*k*Pi/L)^(2*n)/cosh(k*Pi/2)) )} 
for(n=0, 20, print("a("n") = "a(n))) 
 
Thanks, 
     Paul 
 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Some Remarkable Series for A193543(n), A193544(n)
Date: Thu, 30 Aug 2012 03:24:39 GMT

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

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