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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.
15
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
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
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