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%I #40 Sep 08 2022 08:45:51
%S 0,16,416,34096,5794624,1680121936,82501802464,2065646660464,
%T 1739147340740224,210617970218777104,288533264855755545376,
%U 485294472126860897387056,485518650207447822251456
%N a(n) = numerator of (Zeta(2, 1/4) - Zeta(2, n+1/4)), where Zeta is the Hurwitz Zeta function.
%C For A173947/16 see A173949.
%C a(n+1)/A173948(n+1) =: r(n) = (Zeta(2, 1/4) - Zeta(2, n + 5/4)), the partial sum Sum_{k=0..n} 1/(k + 1/4)^2, n >= 0. The limit is Zeta(2, 1/4) = A282823 = 16*A222183. - _Wolfdieter Lang_, Nov 14 2017
%H G. C. Greubel, <a href="/A173947/b173947.txt">Table of n, a(n) for n = 0..250</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function</a>
%F a(n) = numerator of 8*Catalan + Pi^2 - Zeta(2, (4 n + 1)/4), with the Catalan constant given in A006752.
%F a(n) = numerator(r(n)) with r(n) = Zeta(2, 1/4) - Zeta(2, n + 1/4), with the Hurwitz Zeta function (see the name). With Zeta(2, 1/4) = Psi(1, 1/4) = 8*Catalan + Pi^2 this is the preceding formula, where Psi(1, z) is the Trigamma function. - _Wolfdieter Lang_, Nov 14 2017
%p r := n -> Psi(1, 1/4) - Zeta(0, 2, n+1/4):
%p seq(numer(simplify(r(n))), n=0..13); # _Peter Luschny_, Nov 14 2017
%t Table[Numerator[FunctionExpand[8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4]]], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t Numerator[Table[128*n*Sum[(1 + 4*k + 2*n) / ((1 + 4*k)^2*(1 + 4*k + 4*n)^2), {k, 0, Infinity}], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t Numerator[Table[16*Sum[1/(4*k + 1)^2, {k, 0, n - 1} ], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%o (PARI) for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(4*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 22 2018
%o (Magma) [1] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // _G. C. Greubel_, Aug 22 2018
%Y Cf. A006752, A120268, A173945, A173948 (denominators), A173949.
%K frac,nonn,easy
%O 0,2
%A _Artur Jasinski_, Mar 03 2010
%E Name simplified and offset set to 0 by _Peter Luschny_, Nov 14 2017