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A120269
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Numerator of Sum_{k=1..n} 1/(2k-1)^4.
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5
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1, 82, 51331, 123296356, 9988505461, 146251554055126, 4177234784807204311, 4177316109293528392, 348897735816424941428857, 45469045689642442391390873722, 45469276109166591994111574347
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OFFSET
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1,2
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COMMENTS
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a((p-1)/2) is divisible by prime p > 5.
The limit of the rationals r(n) = Sum_{k=1..n} 1/(2k-1)^4, for n -> infinity, is (Pi^4)/96 = (1 - 1/2^4)*zeta(4), which is approximately 1.014678032.
r(n) = (Psi(3, 1/2) - Psi(3, n+1/2))/(3!*2^4) for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(3, 1/2) = 3!*15*zeta(4) = Pi^4. - Jean-François Alcover, Dec 02 2013
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LINKS
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MATHEMATICA
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Numerator[Table[Sum[1/(2k-1)^4, {k, 1, n}], {n, 1, 20}]]
Table[(PolyGamma[3, 1/2] - PolyGamma[3, n + 1/2])/(3!*2^4) // Simplify // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)
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PROG
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(PARI) for(n=1, 20, print1(numerator(sum(k=1, n, 1/(2*k-1)^4)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [Numerator((&+[1/(2*k-1)^4: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
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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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