|
%I #30 Jul 10 2019 03:08:15
%S 4,24,4,80,4,1260,4,1120,4,264,4,12012,4,312,4,38080,4,1220940,4,
%T 83600,4,552,4,78936,4,24,4,30160,4,62300700,4,2360960,4,24,4,
%U 430666236,4,888,4,47986400,4,202145580,4,79120,4,1128,4,126139728,4,264,4,4240,4
%N The denominators of Zagier's modification of the Bernoulli numbers.
%H Robert Israel, <a href="/A216923/b216923.txt">Table of n, a(n) for n = 1..4655</a>
%H M. W. Coffey, V. de Angelis, A. Dixit, V. H. Moll, et al., <a href="https://arxiv.org/abs/1303.6590">The Zagier polynomials. Part II: Arithmetic properties of coefficients</a>, arXiv:1303.6590 [math.NT], 2013.
%H Atul Dixit, Victor H. Moll, Christophe Vignat, <a href="http://arxiv.org/abs/1209.4110">The Zagier modification of Bernoulli numbers and a polynomial extension. Part I</a>, arXiv:1209.4110v1 [math.NT], 2012.
%H D. Zagier, <a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/ModifiedBernoulliNum/fulltext.pdf">A modified Bernoulli number</a>, Nieuw Archief voor Wiskunde, 16:63-72, 1998.
%F a(n) = denominator(sum_{r=0..n} C(n+r,2*r)*B(r)/(n+r)); B(r) the Bernoulli numbers.
%F a(n)=4 if n is odd. - _Robert Israel_, Mar 08 2018
%p f:= proc(n) if n::odd then 4 else denom(-n/4 + add(binomial(n+r,2*r)*bernoulli(r)/(n+r),r=0..n,2)) fi end proc:
%p map(f, [$1..100]); # _Robert Israel_, Mar 08 2018
%t a[n_] := Sum[ Binomial[n + k, 2*k]*BernoulliB[k]/(n + k), {k, 0, n}] // Denominator; Table[a[n], {n, 1, 53}] (* _Jean-François Alcover_, Jul 26 2013 *)
%o (Sage)
%o def A216923(n):
%o return add(binomial(n+r,2*r)*bernoulli(r)/(n+r) for r in (0..n)).denominator()
%o [A216923(n) for n in (1..53)]
%o (PARI) a(n) = denominator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ _Michel Marcus_, Aug 05 2018
%Y Cf. A216922 (numerators).
%K nonn
%O 1,1
%A _Peter Luschny_, Sep 20 2012
|
