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

1, 10, 259, 12916, 117469, 14312974, 2430898831, 487983368, 141433003757, 51174593563322, 51270597630767, 27164483940418988, 3400039831130408821, 30634921277843705014, 25789165074168004597399

Offset: 1

Author

Alexander Adamchuk, Jul 01 2006

Keywords

• nonn
• frac

Comments

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

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.

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-François Alcover, Dec 02 2013 [Corrected by Petros Hadjicostas, May 09 2020]

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..300
• Wolfdieter Lang, Rationals and limit.

Crossrefs

Cf. A007406, A025550, A128492 (denominators).

Programs

Magma

[Numerator((&+[1/(2*k-1)^2: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Mathematica

Numerator[Table[Sum[1/(2k-1)^2,{k,1,n}],{n,1,25}]]
Table[(PolyGamma[1, 1/2] - PolyGamma[1, n+1/2])/4 // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)
Accumulate[1/(2*Range[20]-1)^2]//Numerator (* Harvey P. Dale, Jun 14 2020 *)

PARI

for(n=1,20, print1(numerator(sum(k=1,n, 1/(2*k-1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

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]