A330905 - OEIS

%I #36 May 01 2020 23:05:37

%S 1,13,4009,13739,26190337,790092547807,15121363327643,

%T 2442193001593535677,41701392468830919939353,

%U 17185858614142258665062467,17099921279612182344285033157,28295511786898541163838004665601843,727977487532189566289706245511979571,22681093492188834346091000609641534617709

%N Numerator of 1/Pi^(4*n+3) * Sum_{k>0} (-1)^(k+1) / (k^(4*n+3) * sinh(Pi * k)).

%H Seiichi Manyama, <a href="/A330905/b330905.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 a(n)/A330906(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)!)).

%e 1/360, 13/453600, 4009/13621608000, 13739/4547140416000, 26190337/844351508246400000, 790092547807/2481187700290640140800000, ... = a(n)/A330906(n).

%t a[n_] := Numerator[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, 14, 0] (* _Amiram Eldar_, May 01 2020 *)

%o (PARI) {a(n) = numerator(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, A330906 (denominator).

%K nonn,frac

%O 0,2

%A _Seiichi Manyama_, May 01 2020