W. Lang, Jul 13 2007

Rationals r(n) =  A130553(n)/ A130554(n).

The sequence of rationals r(n):=2*sum(1/(j*binomial(2*j,j)),j=1..n), n>=1, 

tends, in the limit n->infinity, to  

2*Pi*sqrt(3)/9, which is approximately 1.209199577.

The series s:=lim(r(n),n->infty) is similar to  r(2;n):=2*sum(1/(j^2*binomial(2*j,j)),j=1..n), n>=1,

which gives an improved series for 2*Zeta(2)/3 = Pi^2/9. The usual series definition of Zeta(k) (Euler)  
leads to a divergent Zeta(1). The above considered improved series converges.



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

 [1, 7/6, 6/5, 169/140, 1523/1260, 133/110, 72623/60060, 87149/72072, 823077/680680, 15638477/12932920, 46915441/38798760, 13834041/11440660, 224803169/185910725, 6936783521/5736673800, 5587964507/4621209450, 4157445593923/3438179830800, 12472336782289/10314539492400, 170187831339/140744203600, 71785227258967/59365905078480, 153825486983593/127212653739600, 4905323862699739/4056670180362800, 21820233734078929/18045188043682800, 5695081004594650211/4709794079401210800, 594819571590998071/491911826070793128, 312280275085274150713/258253708687166392200]


Numerators  A130553(n), n=1..25:

 [1, 7, 6, 169, 1523, 133, 72623, 87149, 823077, 15638477, 46915441, 13834041, 224803169, 6936783521, 5587964507, 4157445593923, 12472336782289, 170187831339, 71785227258967, 153825486983593, 4905323862699739, 21820233734078929, 5695081004594650211, 594819571590998071, 312280275085274150713].

Denominators  A130554(n), n=1..25:

[1, 6, 5, 140, 1260, 110, 60060, 72072, 680680, 12932920, 38798760, 11440660, 185910725, 5736673800, 4621209450, 3438179830800, 10314539492400, 140744203600, 59365905078480, 127212653739600, 4056670180362800, 18045188043682800, 4709794079401210800, 491911826070793128, 258253708687166392200].



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

[1., 1.209199237, 1.209199576, 1.209199576].


They should be compared with the value for 2*Pi*sqrt(3)/9:

1.209199577. (maple10, 10 digits).


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