%I #27 May 01 2020 23:05:56
%S 360,453600,13621608000,4547140416000,844351508246400000,
%T 2481187700290640140800000,4625642784113264833920000000,
%U 72771380848009396571232614400000000,121040492221732333298138065066291200000000,4859044199288026228257452368062289920000000000
%N Denominator of 1/Pi^(4*n+3) * Sum_{k>0} (-1)^(k+1) / (k^(4*n+3) * sinh(Pi * k)).
%H Seiichi Manyama, <a href="/A330906/b330906.txt">Table of n, a(n) for n = 0..100</a>
%H MathStackexchange, <a href="https://math.stackexchange.com/questions/2598443/a-ramanujan-sum-involving-sinh">A Ramanujan sum involving sinh</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%F Let B_n be the Bernoulli number.
%F A330905(n)/a(n) = Sum_{k=0..2*n+2} (-1)^k*(1-2^(2*k-1))*(1-2^(4*n+3-2*k))*B_{2*k}*B_{4*n+4-2*k}/((2*k)!*(4*n+4-2*k)!)).
%t a[n_] := Denominator[Sum[(-1)^k * (1 - 2^(2*k - 1)) * (1 - 2^(4*n + 3 - 2*k)) * BernoulliB[2*k] * BernoulliB[4*n + 4 - 2*k]/((2*k)!*(4*n + 4 - 2*k)!), {k, 0, 2*n + 2}]]; Array[a, 10, 0] (* _Amiram Eldar_, May 01 2020 *)
%o (PARI) {a(n) = denominator(sum(k=0, 2*n+2, (-1)^k*(1-2^(2*k-1))*(1-2^(4*n+3-2*k))*bernfrac(2*k)*bernfrac(4*n+4-2*k)/((2*k)!*(4*n+4-2*k)!)))}
%Y Cf. A004767, A057866/A057867, A330905 (numerator).
%K nonn,frac
%O 0,1
%A _Seiichi Manyama_, May 01 2020