A360966 - OEIS

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A360966

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.

2

1, 1, 3, 45, 63, 14175, 93555, 42567525, 127702575, 97692469875, 371231385525, 2143861251406875, 2275791174570375, 48076088562799171875, 95646113035463615625, 3952575621190533915703125, 1441527579493018251609375, 68739242628124575327993046875, 333120945043988326589504765625

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OFFSET

0,3

COMMENTS

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.

For numerators see A360945.

LINKS

Table of n, a(n) for n=0..18.

Wikipedia, Hurwitz zeta function

FORMULA

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

(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)!.

EXAMPLE

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

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

MATHEMATICA

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

(* Second program: *)

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

Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023 *)

PROG

(PARI) a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023

CROSSREFS

Bisection of A046983.

Cf. A360945 (numerators).

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

Sequence in context: A119182 A178337 A161589 * A079038 A101790 A124494

Adjacent sequences: A360963 A360964 A360965 * A360967 A360968 A360969

KEYWORD

nonn,frac

AUTHOR

Artur Jasinski, Apr 09 2023

STATUS

approved