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%I #26 Sep 08 2022 08:45:25
%S 1,82,51331,123296356,9988505461,146251554055126,4177234784807204311,
%T 4177316109293528392,348897735816424941428857,
%U 45469045689642442391390873722,45469276109166591994111574347
%N Numerator of Sum_{k=1..n} 1/(2k-1)^4.
%C a((p-1)/2) is divisible by prime p > 5.
%C Denominators are in A128493.
%C 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.
%C 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
%H G. C. Greubel, <a href="/A120269/b120269.txt">Table of n, a(n) for n = 1..293</a>
%H W. Lang, <a href="/A120269/a120269.txt">Rationals and limit</a>.
%t Numerator[Table[Sum[1/(2k-1)^4,{k,1,n}],{n,1,20}]]
%t Table[(PolyGamma[3, 1/2] - PolyGamma[3, n + 1/2])/(3!*2^4) // Simplify // Numerator, {n, 1, 15}] (* _Jean-François Alcover_, Dec 02 2013 *)
%o (PARI) for(n=1,20, print1(numerator(sum(k=1,n, 1/(2*k-1)^4)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o (Magma) [Numerator((&+[1/(2*k-1)^4: k in [1..n]])): n in [1..20]]; // _G. C. Greubel_, Aug 23 2018
%Y Cf. A007410, A013662, A025550.
%K nonn,frac
%O 1,2
%A _Alexander Adamchuk_, Jul 01 2006
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