W. Lang, Jul 13 2007

Rationals A130551(n)/A130552(n)

r(n):=2*sum(((-1)^(j-1))/((j^3)*binomial(2*j,j)),j=1..n), 

tend for n->infinity, to (4/5)*Zeta(3), which is approximately 0.9616455224.

r(n), n=1..25:

 [1, 23/24, 1039/1080, 58157/60480, 1454021/1512000, 6854599/7128000, 30564710941/31783752000, 244517610353/254270016000, 37411196579209/38903312448000, 64619338818497/67196630592000, 86008340157931507/89438715317952000, 8951094220597141/9308101594176000, 334314418075511195849/347648286440879424000, 334314418069194908729/347648286440879424000, 48475590620225838341897/50409001533927516480000, 707173321988579559023843/735378375318472005120000, 3474342530930521624097132659/3612913957939652961154560000, 17696141923244745828114493/18401938665227434437120000, 139220379556912655579302991713189/144773075208599833806424373760000, 139220379556912393017241313604997/144773075208599833806424373760000, 196828812477014155007661811222513/204679175294917006415979287040000, 10671544746036375152619267010346857/11097170938815717695683746562560000, 129840684925024579177555788276800980319/135019278812570837203384144426667520000, 196855231983101780415315949677989021203/204706648522284817695453380259786240000, 152562804786903879982512231966443622141829/158647652604770733713976369701334336000000]


Numerators A130551(n), n=1..25:

[1, 23, 1039, 58157, 1454021, 6854599, 30564710941, 244517610353, 37411196579209, 64619338818497, 86008340157931507, 8951094220597141, 334314418075511195849, 334314418069194908729, 48475590620225838341897, 707173321988579559023843, 3474342530930521624097132659, 17696141923244745828114493, 139220379556912655579302991713189, 139220379556912393017241313604997, 196828812477014155007661811222513, 10671544746036375152619267010346857, 129840684925024579177555788276800980319, 196855231983101780415315949677989021203, 152562804786903879982512231966443622141829]

Denominators A130552(n), n=1..25: 

 [1, 24, 1080, 60480, 1512000, 7128000, 31783752000, 254270016000, 38903312448000, 67196630592000, 89438715317952000, 9308101594176000, 347648286440879424000, 347648286440879424000, 50409001533927516480000, 735378375318472005120000, 3612913957939652961154560000, 18401938665227434437120000, 144773075208599833806424373760000, 144773075208599833806424373760000, 204679175294917006415979287040000, 11097170938815717695683746562560000, 135019278812570837203384144426667520000, 204706648522284817695453380259786240000, 158647652604770733713976369701334336000000]   


The values r(10^k), k=0..3, are (maple10, 10 digits):

[1., .9616455208, .9616455225, .9616455225].   

They should be compared with the value for  (4/5)*Zeta(3):

0.9616455224 (maple10, 10 digits).


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