A216923 The denominators of Zagier's modification of the Bernoulli numbers.

4, 24, 4, 80, 4, 1260, 4, 1120, 4, 264, 4, 12012, 4, 312, 4, 38080, 4, 1220940, 4, 83600, 4, 552, 4, 78936, 4, 24, 4, 30160, 4, 62300700, 4, 2360960, 4, 24, 4, 430666236, 4, 888, 4, 47986400, 4, 202145580, 4, 79120, 4, 1128, 4, 126139728, 4, 264, 4, 4240, 4

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..4655

M. W. Coffey, V. de Angelis, A. Dixit, V. H. Moll, et al., The Zagier polynomials. Part II: Arithmetic properties of coefficients, arXiv:1303.6590 [math.NT], 2013.

Atul Dixit, Victor H. Moll, Christophe Vignat, The Zagier modification of Bernoulli numbers and a polynomial extension. Part I, arXiv:1209.4110v1 [math.NT], 2012.

D. Zagier, A modified Bernoulli number, Nieuw Archief voor Wiskunde, 16:63-72, 1998.

Formula

a(n) = denominator(sum_{r=0..n} C(n+r,2*r)*B(r)/(n+r)); B(r) the Bernoulli numbers.

a(n)=4 if n is odd. - Robert Israel, Mar 08 2018

Maple

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:
map(f, [$1..100]); # Robert Israel, Mar 08 2018

Mathematica

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 *)

Prog

(Sage)
def A216923(n):
    return add(binomial(n+r, 2*r)*bernoulli(r)/(n+r) for r in (0..n)).denominator()
[A216923(n) for n in (1..53)]
(PARI) a(n) = denominator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ Michel Marcus, Aug 05 2018

Crossrefs

Cf. A216922 (numerators).

Sequence in context: A024543 A010294 A350887 * A355992 A233149 A169688

Adjacent sequences: A216920 A216921 A216922 * A216924 A216925 A216926

Keyword

nonn

Author

Peter Luschny, Sep 20 2012

Status

approved