cp's OEIS Frontend

A361007 a(n) = numerator of (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.

%I A361007 #33 Jun 11 2023 12:04:21
%S A361007 0,2,8,64,2176,31744,2830336,178946048,30460116992,839461371904,
%T A361007 232711080902656,39611984424992768,955693069653508096,
%U A361007 1975371841521663868928,1124025625663103358205952,369906947004953565463576576,278846808228005417477465964544
%N A361007 a(n) = numerator of (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.
%C A361007 The function (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) is rational for every positive integer n.
%F A361007 a(n) = numerator( [x^(2*n-1)] tan(2*x) ).
%F A361007 a(n) = numerator( (-1)^(n + 1)*4^(2*n)*(2^(2*n) - 1)*B(2*n)/(2*(2*n)!) ) where B(2*n) are Bernoulli numbers.
%e A361007 tan(2*x) = 2*x + (8/3)*x^3 + (64/15)*x^5 + (2176/315)*x^7 + (31744/2835)*x^9 + ...
%t A361007 Table[(Zeta[2*n, 1/4] + Zeta[2*n, 3/4])/Pi^(2*n), {n, 0, 25}] //
%t A361007   FunctionExpand // Numerator
%t A361007 Table[4^(2 k) (2^(2 k) - 1) (-1)^(k + 1) BernoulliB[2 k]/(2 (2 k)!), {k, 0, 25}] // Numerator
%o A361007 (PARI) my(x='x+O('x^50), v = Vec(tan(2*x)/x)); apply(numerator, vector(#v\2, k, v[2*k-1])) \\ _Michel Marcus_, Apr 09 2023
%Y A361007 Cf. A036279 (denominators).
%Y A361007 Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360945, A360966.
%K A361007 nonn,frac
%O A361007 0,2
%A A361007 _Artur Jasinski_, Mar 15 2023