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A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function. 12
0, 1, 26, 2131, 362164, 105007621, 5156362654, 129102916279, 108696708796264, 13163623138673569, 18033329053484721586, 30330904507928806086691, 30344915637965488890716, 1487479897654682071525709 (list; graph; refs; listen; history; text; internal format)
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 A201623

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

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Last modified April 9 17:48 EDT 2020. Contains 333361 sequences. (Running on oeis4.)