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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.

%I A173949 #42 Feb 16 2025 08:33:12
%S A173949 0,1,26,2131,362164,105007621,5156362654,129102916279,108696708796264,
%T A173949 13163623138673569,18033329053484721586,30330904507928806086691,
%U A173949 30344915637965488890716,1487479897654682071525709
%N A173949 a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4))/16, where Zeta is the Hurwitz Zeta function.
%C A173949 For the Catalan constant see A006752.
%C A173949 The denominators are given in A173948.
%C A173949 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
%H A173949 G. C. Greubel, <a href="/A173949/b173949.txt">Table of n, a(n) for n = 0..250</a>
%H A173949 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%H A173949 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>
%F A173949 a(n) = numerator of expression (8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4))/16.
%F A173949 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
%e A173949 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
%p A173949 r := n -> (Psi(1, 1/4) - Zeta(0, 2, n+1/4))/16:
%p A173949 seq(numer(simplify(r(n))), n=0..13); # _Peter Luschny_, Nov 14 2017
%t A173949 Table[Numerator[FunctionExpand[(8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4])/16]], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t A173949 Numerator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1}], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%o A173949 (PARI) for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o A173949 (Magma) [0] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // _G. C. Greubel_, Aug 23 2018
%Y A173949 Cf. A016814, A120268, A173945, A173947, A173948, A222183.
%K A173949 nonn,frac,easy
%O A173949 0,3
%A A173949 _Artur Jasinski_, Mar 03 2010
%E A173949 Edited by _Wolfdieter Lang_, Nov 14 2017
%E A173949 Name changed according to a formula of Lang by _Peter Luschny_, Nov 14 2017