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A182918 Denominators of the swinging Bernoulli number b_n.

%I A182918 #16 May 08 2020 17:51:01
%S A182918 1,2,6,1,120,1,1512,1,17280,1,190080,1,1415232000,1,21772800,1,
%T A182918 829108224000,1,105082151731200,1,4345502515200000,1,
%U A182918 19989311569920000,1,626378114550988800000,1,17896517558599680000,1,944578196742891110400000
%N A182918 Denominators of the swinging Bernoulli number b_n.
%C A182918 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.
%C A182918 Then 2 zeta(n) [n even] = (2 Pi)^n | B_n | / n! for n >= 2.
%C A182918 Replacing in this formula the factorial of n by the swinging factorial of n (A056040) defines the 'swinging Bernoulli number' b_n.
%C A182918 Then 2 zeta(n) [n even] = (2 Pi)^n b_n / n$ for n >= 2.
%C A182918 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)).
%H A182918 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%e A182918 1, 1/2, 1/6, 0, 1/120, 0, 1/1512, 0, 1/17280, 0, 1/190080, ..
%p A182918 swbern:= proc(n) local swfact;
%p A182918 swfact := n -> n!/iquo(n,2)!^2;
%p A182918 if n=0 then 1 elif n=1 then 1/2 else
%p A182918    if n mod 2 = 1 then 0
%p A182918    else 2*Zeta(n)*swfact(n)/(2*Pi)^n fi
%p A182918 fi end:
%p A182918 Abs_A120082 := n -> numer(swbern(n));
%p A182918 A182918 := n -> denom(swbern(n));
%p A182918 seq(A182918(i),i=0..20);
%t A182918 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 *)
%Y A182918 Cf. A120082.
%K A182918 nonn,frac
%O A182918 0,2
%A A182918 _Peter Luschny_, Feb 03 2011