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A164656 Numerators of partial sums of Theta(5) = sum( 1/(2*j-1)^5, j=1..infinity ).

%I A164656 #17 Aug 29 2019 17:18:46
%S A164656 1,244,762743,12820180976,3115356499043,501734380891571068,
%T A164656 186290962962179367466549,186291207179611798681792,
%U A164656 264507060005034822095008296869,654945930087597102815813733559637156,654946089730308117005814730177159031,4215458332009996232497953858159263996273008
%N A164656 Numerators of partial sums of Theta(5) = sum( 1/(2*j-1)^5, j=1..infinity ).
%C A164656 The denominators are given by A164657.
%C A164656 Rationals (partial sums) Theta(5,n) := sum(1/(2*j-1)^5,j=1..n) (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763.., see A013663.
%C A164656 This is a member of the k-family of rational sequences Theta(k,n):=sum(1/(2*j-1)^k,j=1..n), k>=1, which includes A025550/A025547 (but only for the first 38 entries), A120268/A128492, A164655(n)/A128507(n) (the denominators may depart for higher n values), A120269/A128493, a(n)/A164657, for k=1..5.
%H A164656 R. Ayoub, <a href="http://www.jstor.org/stable/2319041">Euler and the Zeta Function</a>, Am. Math. Monthly 81 (1974) 1067-1086.
%H A164656 W. Lang: <a href="/A164655/a164655.txt">Theta(k,n), k-family of rational sequences and limits.</a>
%F A164656 a(n) = numer(Theta(5,n))= numerator(sum(1/(2*j-1)^5,j=1..n)), n>=1.
%F A164656 Theta(5,n) =  (-Psi(4, 1/2) + Psi(4, n+1/2))/(4!*2^5),  n >= 1, with Psi(n,k) = Polygamma(n,k) is the n^th derivative of the digamma function. Psi(4, 1/2) = -4!*31*Zeta(5). - _Jean-François Alcover_, Dec 02 2013
%e A164656 Rationals Theta(5,n): [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875,...].
%t A164656 r[n_] := Sum[1/(2*j-1)^5, {j, 1, n}]; (* or r[n_] := (PolyGamma[4, n+1/2] - PolyGamma[4, 1/2])/768 // FullSimplify; *) Table[r[n] // Numerator, {n, 1, 12}] (* _Jean-François Alcover_, Dec 02 2013 *)
%K A164656 nonn,frac,easy
%O A164656 1,2
%A A164656 _Wolfdieter Lang_, Oct 16 2009