A182918 Denominators of the swinging Bernoulli number b_n.

1, 2, 6, 1, 120, 1, 1512, 1, 17280, 1, 190080, 1, 1415232000, 1, 21772800, 1, 829108224000, 1, 105082151731200, 1, 4345502515200000, 1, 19989311569920000, 1, 626378114550988800000, 1, 17896517558599680000, 1, 944578196742891110400000

Offset

0,2

Comments

Let zeta(n) denote the Riemann zeta function, B_n the Bernoulli numbers and let [n even] be 1 if n is even, 0 otherwise.

Then 2 zeta(n) [n even] = (2 Pi)^n | B_n | / n! for n >= 2.

Replacing in this formula the factorial of n by the swinging factorial of n (A056040) defines the 'swinging Bernoulli number' b_n.

Then 2 zeta(n) [n even] = (2 Pi)^n b_n / n$ for n >= 2.

Let additionally b_0 = 1 and b_1 = 1/2. The b_n are rational numbers like the Bernoulli numbers; unlike the Bernoulli numbers the swinging Bernoulli numbers are unsigned, bounded in the interval [0,1] and approach 0 for n -> infinity. The numerators of the swinging Bernoulli numbers b_n are abs(A120082(n)).

Links

Table of n, a(n) for n=0..28.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Example

1, 1/2, 1/6, 0, 1/120, 0, 1/1512, 0, 1/17280, 0, 1/190080, ..

Maple

swbern:= proc(n) local swfact;
swfact := n -> n!/iquo(n, 2)!^2;
if n=0 then 1 elif n=1 then 1/2 else
   if n mod 2 = 1 then 0
   else 2*Zeta(n)*swfact(n)/(2*Pi)^n fi
fi end:
Abs_A120082 := n -> numer(swbern(n));
A182918 := n -> denom(swbern(n));
seq(A182918(i), i=0..20);

Mathematica

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[1] = 2; a[_?OddQ] = 1; a[n_] := 2*Zeta[n]*sf[n]/(2*Pi)^n // Denominator; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jul 26 2013 *)

Crossrefs

Cf. A120082.

Sequence in context: A347427 A117214 A185972 * A134301 A168294 A363737

Adjacent sequences: A182915 A182916 A182917 * A182919 A182920 A182921

Keyword

nonn,frac

Author

Peter Luschny, Feb 03 2011

Status

approved