A120269 numerator of rationals r(n):=sum(1/(2*k-1)^4,k=1..infty).

The rationals r(n) for n=1..21 are

[1, 82/81, 51331/50625, 123296356/121550625, 9988505461/9845600625, 
146251554055126/144149438750625, 4177234784807204311/4117052120156600625, 
4177316109293528392/4117052120156600625, 
348897735816424941428857/343860310127599440800625, 
45469045689642442391390873722/44812219476138886724578250625, 
45469276109166591994111574347/44812219476138886724578250625, 
12724212507884764409110901655089452/12540296310422182199892702233150625, 
7952652881902074431185833362754480541/7837685194013863874932938895719140625, 
644166078021050808054824782971269564446/634852500715122973869568050553250390625, 
455607062681307551694798396892753062061321951/449019111558292892081441960363353489531640625, 

420763139153615389741623991738752563981423641150096/
414678578925426209987945360676724578005750281640625, 

420763488821577189411424921096914428662610672790721/
414678578925426209987945360676724578005750281640625, 

60109109308403231537976272759342454095213484534478/
59239796989346601426849337239532082572250040234375, 

112654207650343298169046576180259252641621971638662057333/
111024917165350815876745380730178687405690707655696484375, 

112654255641589742470334517712107581679919211849546041708/
111024917165350815876745380730178687405690707655696484375, 

318334113100951437623530813850063562294117597685892728059324163
/313729880954078886822687903797490457902191979755868553383984375]


The numerators A120269(n) for n=1..21 are:

[1, 82, 51331, 123296356, 9988505461, 146251554055126, 4177234784807204311, 4177316109293528392, 348897735816424941428857, 45469045689642442391390873722, 45469276109166591994111574347, 12724212507884764409110901655089452, 7952652881902074431185833362754480541, 644166078021050808054824782971269564446, 455607062681307551694798396892753062061321951, 420763139153615389741623991738752563981423641150096, 420763488821577189411424921096914428662610672790721, 60109109308403231537976272759342454095213484534478, 112654207650343298169046576180259252641621971638662057333, 112654255641589742470334517712107581679919211849546041708, 318334113100951437623530813850063562294117597685892728059324163] 


The denominators A128493(n) for n=1..21 are:

[1, 81, 50625, 121550625, 9845600625, 144149438750625, 4117052120156600625, 4117052120156600625, 343860310127599440800625, 44812219476138886724578250625, 44812219476138886724578250625, 12540296310422182199892702233150625, 7837685194013863874932938895719140625, 634852500715122973869568050553250390625, 449019111558292892081441960363353489531640625, 414678578925426209987945360676724578005750281640625, 414678578925426209987945360676724578005750281640625, 59239796989346601426849337239532082572250040234375, 111024917165350815876745380730178687405690707655696484375, 111024917165350815876745380730178687405690707655696484375, 313729880954078886822687903797490457902191979755868553383984375]


The values r(n) for n=10^k-1, k=0..5 are:

[1.012345679, 1.014662443, 1.014678011, 1.014678032, 1.014678032, 1.014678032]

This should be compared with the approximate value for (Pi^4)/96 (maple10, 10 digits)

1.014678032.

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Note: sum(1/(2*k+1)^4,k=0..infinity) is called Theta(4) and equals (1-1/2^4)*Zeta(4) with 
Zeta(4):=sum(1/k^4,k=1..infinity)= (Pi^4)/90. See the rationals A007410/A007480.    
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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