W. Lang, Jul 13 2007

Rationals r(n) = A130557(n)/A130558(n).

r(n):= 4!*sum(1/((j^3)*(j^2-1)),j=2..n), n>=2, tends, 

in the limit n->infinity, to 

6*(5-4*Zeta(3)) which is approximately 1.15063433.


r(n), n=2..25:

[1, 10/9, 409/360, 10297/9000, 8031/7000, 394019/343000, 9462581/8232000, 766743461/666792000, 8435956183/7334712000, 1020884056543/887500152000, 13272613316059/11537501976000, 2243198436149971/1949837833944000, 2243285892433171/1949837833944000, 2243347792046947/1949837833944000, 305101392961615867/265177945416384000, 88175602457796281563/76636426225334976000, 186150555360181760633/161788010920151616000, 67200918162256914366113/58405471942174733376000, 13440271460228739526957/11681094388434946675200, 13440340259559589925677/11681094388434946675200, 309129079709742676796171/268665170934003773529600, 163529814125289713977296059/142123875424087996197158400, 163530243243271066899989891/142123875424087996197158400, 102206620679160453870949369811/88827422140054997623224000000].



Numerators A130557(n), n=2..25:

[1, 10, 409, 10297, 8031, 394019, 9462581, 766743461, 8435956183, 1020884056543, 13272613316059, 2243198436149971, 2243285892433171, 2243347792046947, 305101392961615867, 88175602457796281563, 186150555360181760633, 67200918162256914366113, 13440271460228739526957, 13440340259559589925677, 309129079709742676796171, 163529814125289713977296059, 163530243243271066899989891, 102206620679160453870949369811]



Denominators A130558(n), n=2..25:

[1, 9, 360, 9000, 7000, 343000, 8232000, 666792000, 7334712000, 887500152000, 11537501976000, 1949837833944000, 1949837833944000, 1949837833944000, 265177945416384000, 76636426225334976000, 161788010920151616000, 58405471942174733376000, 11681094388434946675200, 11681094388434946675200, 268665170934003773529600, 142123875424087996197158400, 142123875424087996197158400, 88827422140054997623224000000]


The values r(10^k), k=1..4, are (maple10, 10 digits):

[1.150141435, 1.150634265, 1.150634324, 1.15063433].

They should be compared with the value for 6*(5-4*Zeta(3)) = 30*(1-(4/5)*Zeta(3)):

1.15063433 (maple10, 10 digits).


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