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

0,3

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}_{k>=0} see A016814. The limit n -> infinity is given in A222183 as 1.074833072... . - 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

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

Example

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

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 *)

Prog

(PARI) for(n=0, 20, print1(numerator(sum(k=0, n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Crossrefs

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

Sequence in context: A245929 A206390 A268089 * A266030 A103642 A363990

Adjacent sequences: A173946 A173947 A173948 * A173950 A173951 A173952

Keyword

nonn,frac,easy

Author

Artur Jasinski, Mar 03 2010

Extensions

Edited by Wolfdieter Lang, Nov 14 2017

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

Status

approved