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Denominators of partial sums of Theta(5) = Sum_{j>=1} 1/(2*j-1)^5.
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%I #19 Oct 07 2024 06:32:13

%S 1,243,759375,12762815625,3101364196875,499477805270915625,

%T 185452612752454075153125,185452612752454075153125,

%U 263316190384861185784690603125,651996955695764397260286617707209375,651996955695764397260286617707209375,4196476041813743307955464949873473110315625

%N Denominators of partial sums of Theta(5) = Sum_{j>=1} 1/(2*j-1)^5.

%C The numerators are given by A164656.

%C For a reference and a W. Lang link see A164656.

%C Rationals (partial sums) Theta(5,n) := Sum_{j=1..n} 1/(2*j-1)^5 (in lowest terms). The limit of these rationals is Theta(5)= (1-1/2^5)*Zeta(5) approximately 1.004523763 (Zeta(n) is the Euler-Riemann zeta function).

%F a(n) = denominator(Theta(5,n)) = denominator(Sum_{j=1..n} 1/(2*j-1)^5).

%e Rationals Theta(5,n): [1, 244/243, 762743/759375, 12820180976/12762815625, 3115356499043/3101364196875,...].

%t 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] // Denominator, {n, 1, 12}] (* _Jean-François Alcover_, Dec 02 2013 *)

%K nonn,frac,easy

%O 1,2

%A _Wolfdieter Lang_, Oct 16 2009