W. Lang, Aug 07 2007

Rationals r(n)=A131447(n)/A131448(n).

r(n):=sum(A096622(k)/k!,n=1..n) with A096622(k) the factorial expansion of the Euler 

constant gamma.


 A096622(k), k=1..50:

[0, 1, 0, 1, 4, 1, 4, 1, 3, 0, 2, 3, 0, 5, 14, 12, 16, 14, 7, 13, 18, 21, 17, 15, 24, 5, 21, 19, 10, 16, 4, 20, 12, 27, 16, 20, 23, 15, 17, 36, 5, 4, 14, 33, 18, 18, 30, 26, 1, 17,...]



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

 [0, 1/2, 1/2, 13/24, 23/40, 83/144, 2909/5040, 23273/40320, 3491/6048, 3491/6048, 11520301/19958400, 30720803/53222400, 30720803/53222400, 50320675319/87178291200, 68619102709/118879488000, 3019240519199/5230697472000, 4666098984217/8083805184000, 1847775197749939/3201186852864000, 23405152504832563/40548366802944000, 1404309150289953793/2432902008176640000, 9830164052029676557/17030314057236480000, 92684403919136950397/160571532539658240000, 932636814436315563371/1615751046180311040000, 1101946266903215927183/1909072005333044428800, 426348258028029971826757/738629049682427904000000,...].


The values r(10^k), k=0..3, are (maple11, 10 digits):

[0., .5772156085, .5772156649, .5772156649].

This should be compared with gamma (maple11, 10 digits):

0.5772156649 .


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