A216922 The numerators of Zagier's modification of the Bernoulli numbers.

3, 1, -1, -27, -1, -29, 1, 451, 1, -65, -3, -6571, 3, 571, -1, -181613, -1, 23663513, 1, -10188203, 1, 564133, -3, -854671223, 3, 3380293, -1, -66346796677, -1, 2525207721201139, 1, -2050779016779123, 1, 513555084737, -3, -258395660795799074117, 3

Offset

1,1

Links

Seiichi Manyama, Table of n, a(n) for n = 1..629

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

Mathematica

a[n_] := Sum[ Binomial[n + k, 2*k]*BernoulliB[k]/(n + k), {k, 0, n}] // Numerator; Table[a[n], {n, 1, 37}] (* Jean-François Alcover, Jul 26 2013 *)

Prog

(SageMath)
def A216922(n):
    return add(binomial(n+r, 2*r)*bernoulli(r)/(n+r) for r in (0..n)).numerator()
[A216922(n) for n in (1..37)]
(PARI) a(n) = numerator(sum(r=0, n, binomial(n+r, 2*r)*bernfrac(r)/(n+r))); \\ Michel Marcus, Aug 05 2018

Crossrefs

Cf. A216923 (denominators).

Sequence in context: A156950 A083998 A277170 * A245243 A168242 A332540

Adjacent sequences: A216919 A216920 A216921 * A216923 A216924 A216925

Keyword

sign

Author

Peter Luschny, Sep 20 2012

Status

approved