W. Lang, Oct 17 2008

Rationals r(n) = A145564(n)/A145565(n) =: sum(1/(binomial(2*k,k)*(k+1)),k=0..n),n>=0,
(in lowest terms). 


Numerators are A145564(n), n=0..25:

[1, 5, 47, 949, 33287, 14273, 7694047, 400101469, 1200312247, 20405339951, 4264717637359, 
328055232193, 1275150714976991, 1275150721602467, 2125251205342781, 246529139894912671, 
129920856734238187217, 2122119257040297503, 22216466502052353380347, 164401852115363364287267,
 164401852115406331142627, 6740475936732090221920747, 6666330701428141445587294463, 
1333266140285633393776807081, 7832938574178103539148202992171, 602533736475238872477449420167,...].


Denominators are A145565(n), n=0..25:

[1, 4, 36, 720, 25200, 10800, 5821200, 302702400, 908107200, 15437822400, 3226504881600, 248192683200,
 964724959598400, 964724959598400, 1607874932664000, 186513492189024000, 98292610383615648000, 
1605505432763232000, 16808036375598275808000, 124379469179427240979200, 124379469179427240979200, 
5099558236356516880147200, 5043463095756595194465580800, 1008692619151319038893116160, 
5926069137513999353497057440000, 455851472116461488730542880000,...].



The rationals r(n) are, for n=0..25:

[1, 5/4, 47/36, 949/720, 33287/25200, 14273/10800, 7694047/5821200, 400101469/302702400,
 1200312247/908107200, 20405339951/15437822400, 4264717637359/3226504881600, 328055232193/248192683200, 
1275150714976991/964724959598400, 1275150721602467/964724959598400, 2125251205342781/1607874932664000,
 246529139894912671/186513492189024000, 129920856734238187217/98292610383615648000,
 2122119257040297503/1605505432763232000, 22216466502052353380347/16808036375598275808000,
 164401852115363364287267/124379469179427240979200, 164401852115406331142627/124379469179427240979200,
 6740475936732090221920747/5099558236356516880147200,
 6666330701428141445587294463/5043463095756595194465580800,
 1333266140285633393776807081/1008692619151319038893116160,
 7832938574178103539148202992171/5926069137513999353497057440000,
 602533736475238872477449420167/455851472116461488730542880000]


Some values are: r(10^k), k=0..3: [1.250000000, 1.321776285, 1.321776441, 1.321776441] (Maple11, 10 digits).

This should be compared with the limit (n -> infinity) of r(n): (4*sqrt(3)- Pi)*Pi/9. 
This limit is approximately 1.321776442 (Maple11, 10 digits). 


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