W. Lang, Jul 13 2007

Rationals r(n)= A130549(n)/A130550(n)

The sequence r(n):= 2*sum(1/(j^2*binomial(2*j,j)),j=1..n) tends, for n->infinity, 

to 2*Zeta(2)/3 = (Pi^2)/9, which is approximately 1.096622711.    

See the A. van der Poorten reference given in  A130549.


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

[1, 13/12, 197/180, 1105/1008, 9211/8400, 130277/118800, 82987349/75675600, 331950131/302702400, 16929464521/15437822400, 29241805241/26665329600, 3538258509761/3226504881600, 6259995854281/5708431713600, 1057939300471201/964724959598400, 1057939300716589/964724959598400, 51133732870640471/46628373047256000, 372975463296151087/340112838697632000, 107789908892879155343/98292610383615648000, 51058377896658637853/46559657550133728000, 681986753565766904623961/621897345897136204896000, 27279470142631578488921/24875893835885448195840]


Numerators A130549(n), n=1..25:

 [1, 13, 197, 1105, 9211, 130277, 82987349, 331950131, 16929464521, 29241805241, 3538258509761, 6259995854281, 1057939300471201, 1057939300716589, 51133732870640471, 372975463296151087, 107789908892879155343, 51058377896658637853, 681986753565766904623961, 27279470142631578488921]


Denominators A130550(n), n=1..25:
 
[1, 12, 180, 1008, 8400, 118800, 75675600, 302702400, 15437822400, 26665329600, 3226504881600, 5708431713600, 964724959598400, 964724959598400, 46628373047256000, 340112838697632000, 98292610383615648000, 46559657550133728000, 621897345897136204896000, 24875893835885448195840]



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

[1., 1.096622681, 1.096622711, 1.096622711]

They shouls be compared with the value for  2*Zeta(2)/3 = (Pi^2)/9:

1.096622711 (maple10, 10 digits).


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