A120268 numerators of rationals

r(n):=sum(1/(2*k-1)^2,k=1..n), n>=1.

R(n) for n=1..21 is:

[1, 10/9, 259/225, 12916/11025, 117469/99225, 14312974/12006225, 2430898831/2029052025, 487983368/405810405, 141433003757/117279207045, 51174593563322/42337793743245, 51270597630767/42337793743245, 27164483940418988/22396692890176605, 3400039831130408821/2799586611272075625, 30634921277843705014/25196279501448680625, 25789165074168004597399/21190071060718340405625, 24804577707336170758506064/20363658289350325129805625, 24823277118070193095631689/20363658289350325129805625, 3548557216084587389770102/2909094041335760732829375, 4860883922861135897328099013/3982549742588656443243414375, 4863502298760273738513663388/3982549742588656443243414375, 8179529913958608810884711569603/6694666117291531481092179564375]


The numerators A120268(n) for n=1..21 are:

 [1, 10, 259, 12916, 117469, 14312974, 2430898831, 487983368, 141433003757, 51174593563322, 51270597630767, 27164483940418988, 3400039831130408821, 30634921277843705014, 25789165074168004597399, 24804577707336170758506064, 24823277118070193095631689, 3548557216084587389770102, 4860883922861135897328099013, 4863502298760273738513663388, 8179529913958608810884711569603]
 

The denominators A1284928(n) for n=1..21 are:

 [1, 9, 225, 11025, 99225, 12006225, 2029052025, 405810405, 117279207045, 42337793743245, 42337793743245, 22396692890176605, 2799586611272075625, 25196279501448680625, 21190071060718340405625, 20363658289350325129805625, 20363658289350325129805625, 2909094041335760732829375, 3982549742588656443243414375, 3982549742588656443243414375, 6694666117291531481092179564375]


The rationals for the values n=10^k-1, k=0..5 are:

[1.111111111, 1.210988885, 1.231225323, 1.233450800, 1.233675552, 1.233698050]   

This should be compared to the limit (Pi^2)/8 which is approximately (maple10, 10 digits):

 1.233700550.

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Note: sum(1/(2*k+1)^2,k=0..infinity) is called Theat(2) and equals (1-1/2^2)*Zeta(2) with 
Zeta(2):=sum(1/k^2,k=1..infinity)= (Pi^2)/6. See the rationals A007406(n)/A007407(n).    
For Euler and the Zeta function see the R. Ayoub reference given in A127676.
The sum here is a special case of the Theta(s):=sum(1/(2*k+1)^s,k=0..infty) functions 
which converge for s>1.  

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