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A281820
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Numerator of Sum_{k=1..n} (30k-11)/(4*(2k-1)*k^3*binomial(2k,k)^2).
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2
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19, 4153, 519283, 1424927267, 38473051777, 51207632802437, 112503169355608589, 7200202839028523, 884364913705304409923, 30329294715526225502633653, 30329294715526370166581653, 369016528803809437978645999301
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OFFSET
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1,1
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COMMENTS
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In 1990, Gosper gave the following combinatorial identity: zeta(3) = Sum_{k>=1} (30k-11)/(4*(2k-1)*k^3*binomial(2k,k)^2).
Conjecture: Sum_{n >= k+1} 1/(n^3*(n^2 - 1)^2*(n^2 - 4)^2*...*(n^2 - k^2)^2) = Sum_{n >= k+1} 1/(n*binomial(n,k)^2*binomial(n+k,k)^2*(n-k)^2) = zeta(3) - A281820(k)/A281821(k). - Peter Bala, Jan 17 2022
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REFERENCES
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Ralph William Gosper Jr, A calculus of series rearrangements in Algorithms and Complexity, New directions and Recent Results, ed. J. F. Traub, Academic Press Inc., 1976, p. 122.
Lloyd James Peter Kilford, Modular Forms: A Classical and Computational Introduction, World Scientific, 2008 page 188.
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LINKS
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EXAMPLE
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19/16, 4153/3456, 519283/432000, 1424927267/1185408000, ...
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MATHEMATICA
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Table[Sum[(30k-11)/(4(2k-1)k^3 Binomial[2k, k]^2), {k, n}], {n, 20}]//Numerator (* Harvey P. Dale, Dec 31 2021 *)
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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