A360966 - OEIS

%I #29 Jun 11 2023 12:04:02

%S 1,1,3,45,63,14175,93555,42567525,127702575,97692469875,371231385525,

%T 2143861251406875,2275791174570375,48076088562799171875,

%U 95646113035463615625,3952575621190533915703125,1441527579493018251609375,68739242628124575327993046875,333120945043988326589504765625

%N a(n) = denominator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.

%C The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n.

%C For numerators see A360945.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hurwitz_zeta_function">Hurwitz zeta function</a>

%F a(n) = A046983(2*n).

%F (Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!.

%e a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1.

%e a(3) = 45 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.

%t Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 0, 25}] // FunctionExpand // Denominator

%t (* Second program: *)

%t a[n_] := SeriesCoefficient[Tan[x + Pi/4], {x, 0, 2n}] // Denominator;

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 16 2023 *)

%o (PARI) a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ _Michel Marcus_, Apr 11 2023

%Y Bisection of A046983.

%Y Cf. A360945 (numerators).

%Y Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A361007.

%K nonn,frac

%O 0,3

%A _Artur Jasinski_, Apr 09 2023