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%I #45 Sep 08 2022 08:45:51
%S 0,1,58,7459,192404,70791869,37930481726,3100675399831,
%T 3000384410275816,3016572632600497,512004171837010018,
%U 950047080453398607307,2104850677799349861903388,609822785846772474028096357,611130542819711220012487366
%N a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4))/16 where Zeta(n, a) is the Hurwitz Zeta function.
%C The denominators are given in A173954.
%C a(n+2)/A173954(n+2) = (Zeta(2, 3/4) - Zeta(2, n + 7/4))/16 gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(4*n + 3). In the limit n -> infinity the series value is Zeta(2,3/4)/16, with the Hurwitz Zeta function, and it is given in A247037. - _Wolfdieter Lang_, Nov 15 2017
%H G. C. Greubel, <a href="/A173955/b173955.txt">Table of n, a(n) for n = 1..250</a>
%F a(n) = numerator of r(n) with r(n) = (Pi^2 - 8*Catalan - Zeta(2, n - 1/4))/16, with the Hurwitz Zeta function Z(2, z), and the Catalan constant is given in A006752. With Zeta(2, 3/4) = Pi^2 - 8*Catalan this is the formula given in the name.
%F Numerator of Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. - _G. C. Greubel_, Aug 23 2018
%p r := n -> (Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4))/16:
%p seq(numer(simplify(r(n))), n=1..15); # _Peter Luschny_, Nov 14 2017
%t Table[Numerator[FunctionExpand[(Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4])/16]], {n, 1, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t Numerator[Table[8*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t Numerator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-2}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 15 2017 *)
%o (PARI) for(n=1,20, print1(numerator(sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o (Magma) [0] cat [Numerator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // _G. C. Greubel_, Aug 23 2018
%Y Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173954, A247037.
%K frac,nonn,easy
%O 1,3
%A _Artur Jasinski_, Mar 03 2010
%E Numbers changed according to the old (or new) Mathematica program, and edited by _Wolfdieter Lang_, Nov 14 2017
%E Name simplified by _Peter Luschny_, Nov 14 2017
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