This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A173953 #46 Feb 16 2025 08:33:12
%S A173953 0,16,928,119344,3078464,1132669904,606887707616,49610806397296,
%T A173953 48006150564413056,48265162121607952,8192066749392160288,
%U A173953 15200753287254377716912,33677610844789597790454208
%N A173953 a(n) = numerator of (Zeta(2, 3/4) - Zeta(2, n-1/4)), where Zeta is the Hurwitz Zeta function.
%C A173953 All numbers in this sequence are divisible by 16. For A173953/16 see A173955.
%C A173953 a(n+2)/A173954(n+2) is, for n >= 0, the partial sum Sum_{k=0..n} 1/(k + 3/4)^2 = 16*Sum_{k=0..n} 1/(4*k + 3)^2. The limit n -> infinity is given in A282824 as Zeta(2, 3/4) = Psi(1, 3/4) = Pi^2 - 8*Catalan, with the trigamma function Psi(1, z) and the Catalan constant A006752.
%H A173953 G. C. Greubel, <a href="/A173953/b173953.txt">Table of n, a(n) for n = 1..250</a>
%H A173953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HurwitzZetaFunction.html">Hurwitz Zeta Function</a>
%H A173953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrigammaFunction.html">Trigamma Function</a>
%F A173953 a(n) = Numerator of (Pi^2 - 8*Catalan - Zeta(2, (4 n - 1)/4)).
%F A173953 Numerator of 128*n*Sum_{k>=1} (4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2). - _Vaclav Kotesovec_, Nov 14 2017
%F A173953 Numerator of 16*Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. See a comment above. - _Wolfdieter Lang_, Nov 14 2017
%e A173953 The rationals r(n) = Zeta(2, 3/4) - Zeta(2, n-1/4) begin: 0/1, 16/9, 928/441, 119344/53361, 3078464/1334025, 1132669904/481583025, 606887707616/254757420225, 49610806397296/20635351038225, ... - _Wolfdieter Lang_, Nov 14 2017
%p A173953 r := n -> Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4):
%p A173953 seq(numer(simplify(r(n))), n=1..13); # _Peter Luschny_, Nov 14 2017
%t A173953 Table[Numerator[FunctionExpand[Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4]]], {n, 1, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t A173953 Numerator[Table[128*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 A173953 Numerator[Table[16*Sum[1/(4*k + 3)^2, {k, 0, n-1}], {n, 1, 20}]] (* _Vaclav Kotesovec_, Nov 15 2017 *)
%o A173953 (PARI) for(n=1,20, print1(numerator(16*sum(k=0,n-2, 1/(4*k+3)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o A173953 (Magma) [0] cat [Numerator((&+[16/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // _G. C. Greubel_, Aug 23 2018
%Y A173953 Denominators are in A173954.
%Y A173953 Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173955.
%K A173953 frac,nonn,easy
%O A173953 1,2
%A A173953 _Artur Jasinski_, Mar 03 2010
%E A173953 Name simplified by _Peter Luschny_, Nov 14 2017