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A120268 Numerator of Sum_{k=1..n} 1/(2*k-1)^2. 18

%I

%S 1,10,259,12916,117469,14312974,2430898831,487983368,141433003757,

%T 51174593563322,51270597630767,27164483940418988,3400039831130408821,

%U 30634921277843705014,25789165074168004597399

%N Numerator of Sum_{k=1..n} 1/(2*k-1)^2.

%C a((p-1)/2) is divisible by prime p > 3.

%C The limit of the rationals r(n) = Sum_{k=1..n} 1/(2*k-1)^2, for n -> infinity, is (Pi^2)/8 = (1 - 1/2^2)*Zeta(2), which is approximately 1.233700550.

%C r(n) = (Psi(1, 1/2) - Psi(1, n+1/2))/4 for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(1, 1/2) = 3*Zeta(2) = Pi^2/2. - _Jean-Fran├žois Alcover_, Dec 02 2013 [Corrected by _Petros Hadjicostas_, May 09 2020]

%H G. C. Greubel, <a href="/A120268/b120268.txt">Table of n, a(n) for n = 1..300</a>

%H Wolfdieter Lang, <a href="/A120268/a120268.txt">Rationals and limit</a>.

%F a(n) = numerator( Pi^2/2 - Zeta(2,(2n+1)/2) ) / 4 for n > 0. - _Artur Jasinski_, Mar 03 2010 [corrected by _Bruno Berselli_, Dec 02 2013]

%e Fractions begin: 1, 10/9, 259/225, 12916/11025, 117469/99225, 14312974/12006225, 2430898831/2029052025, 487983368/405810405, ... = A120268/A128492.

%t Numerator[Table[Sum[1/(2k-1)^2,{k,1,n}],{n,1,25}]]

%t Table[(PolyGamma[1, 1/2] - PolyGamma[1, n+1/2])/4 // Numerator, {n, 1, 15}] (* _Jean-Fran├žois Alcover_, Dec 02 2013 *)

%t Accumulate[1/(2*Range[20]-1)^2]//Numerator (* _Harvey P. Dale_, Jun 14 2020 *)

%o (PARI) for(n=1,20, print1(numerator(sum(k=1,n, 1/(2*k-1)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018

%o (MAGMA) [Numerator((&+[1/(2*k-1)^2: k in [1..n]])): n in [1..20]]; // _G. C. Greubel_, Aug 23 2018

%Y Cf. A007406, A025550, A128492 (denominators).

%K nonn,frac

%O 1,2

%A _Alexander Adamchuk_, Jul 01 2006

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Last modified June 25 03:53 EDT 2021. Contains 345449 sequences. (Running on oeis4.)