A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.

0, 1, 26, 2131, 362164, 105007621, 5156362654, 129102916279, 108696708796264, 13163623138673569, 18033329053484721586, 30330904507928806086691, 30344915637965488890716, 1487479897654682071525709

Offset: 0

Author

Artur Jasinski, Mar 03 2010

Keywords

• nonn
• frac
• easy

Comments

For the Catalan constant see A006752.

The denominators are given in A173948.

a(n+1)/A173948(n+1), for n>= 0, gives the partial sum Sum_{k=0..n} 1/(4*k + 1)^2. For {(4*k + 1)^2}A016814.%20The%20limit%20n%20-%3E%20infinity%20is%20given%20in%20A222183%20as%201.074833072...%20.%20-%20_Wolfdieter%20Lang">{k>=0} see A016814. The limit n -> infinity is given in A222183 as 1.074833072... . - _Wolfdieter Lang, Nov 14 2017

Examples

			The rationals r(n) begin: 0/1, 1/1, 26/25, 2131/2025, 362164/342225, 105007621/98903025, 5156362654/4846248225, 129102916279/121156205625, 108696708796264/101892368930625, 13163623138673569/12328976640605625, ... - _Wolfdieter Lang_, Nov 14 2017
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
• Eric Weisstein's World of Mathematics, Polygamma Function

Crossrefs

Cf. A016814, A120268, A173945, A173947, A173948, A222183.

Programs

Magma

[0] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Maple

r := n -> (Psi(1, 1/4) - Zeta(0, 2, n+1/4))/16:
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017

Mathematica

Table[Numerator[FunctionExpand[(8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4])/16]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)

PARI

for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018

Formula

a(n) = numerator of expression (8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4))/16.

a(n) = numerator(r(n)) with r(n) = (Zeta(2,1/4) - Zeta(2, n + 1/4))/16, with the Hurwitz Zeta function Z(2, k). With Zeta(2,1/4) = 8 Catalan + Pi^2 this is the preceding formula, and Zeta(2, n + 1/4) = Psi(1, n + 1/4) with the polygamma (trigamma) function Psi(1, k). - Wolfdieter Lang, Nov 14 2017

Extensions

Edited by Wolfdieter Lang, Nov 14 2017

Name changed according to a formula of Lang by Peter Luschny, Nov 14 2017