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A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.

%I A173984 #26 Sep 08 2022 08:45:51
%S A173984 1,1,16,784,19600,3312400,52998400,19132422400,2315023110400,
%T A173984 57875577760000,57875577760000,55618430227360000,16073726335707040000,
%U A173984 22004931353582937760000,22004931353582937760000
%N A173984 a(n) is the denominator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)) where Zeta is the Hurwitz Zeta function.
%H A173984 G. C. Greubel, <a href="/A173984/b173984.txt">Table of n, a(n) for n = 0..300</a>
%F A173984 a(n) = denominator of 2*(Pi^2)/3 + J - Zeta(2,(3*n+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
%F A173984 a(n) = denominator of Sum_{k=1..(n-1)} 9/(3*k+1)^2. - _G. C. Greubel_, Aug 24 2018
%p A173984 a := n -> Zeta(0,2,1/3) - Zeta(0,2,n+1/3):
%p A173984 seq(denom(a(n)), n=0..14); # _Peter Luschny_, Nov 14 2017
%t A173984 Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Denominator (* _Vaclav Kotesovec_, Nov 13 2017 *)
%t A173984 Denominator[Table[Sum[9/(3*k + 1)^2, {k, 1, n - 1}], {n, 0, 30}]] (* _G. C. Greubel_, Aug 24 2018 *)
%o A173984 (PARI) for(n=0,20, print1(denominator(sum(k=1,n-1, 9/(3*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 24 2018
%o A173984 (Magma) [1,1] cat [Denominator((&+[9/(3*k+1)^2: k in [1..n-1]])): n in [2..20]]; // _G. C. Greubel_, Aug 24 2018
%Y A173984 For the numerators see A173982.
%Y A173984 Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173983, A173985, A173986, A173987.
%K A173984 frac,nonn
%O A173984 0,3
%A A173984 _Artur Jasinski_, Mar 04 2010
%E A173984 Name simplified by _Peter Luschny_, Nov 14 2017