OFFSET
0,3
REFERENCES
R. Edwards and D. J. Leeming, The exact number of real roots of the Bernoulli polynomial, Journal of Approximation Theory 164:5 (2012), pp. 754-775.
A. P. Veselov and J. P. Ward, On the real zeros of the Hurwitz zeta-function and Bernoulli polynomials. J. Math. Anal. Appl. 305 (2005), no. 2, 712-721.
LINKS
A. P. Veselov and J. P. Ward, On the real roots of the Bernoulli polynomials and the Hurwitz zeta-function, 1999 preprint.
FORMULA
a(n) = 2n/(Pi*e) + O(log n).
MATHEMATICA
a[n_] := CountRoots[ BernoulliB[n, x], x]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 13 2012 *)
PROG
(PARI) a(n)=polsturm(sum(i=0, n, binomial(n, i)*bernfrac(i)*x^(n-i)))
(PARI) a(n)=my(e=1e-29, v=polroots(bernpol(n))); sum(i=1, #v, abs(imag(v[i])) <= abs(v[i])*e) \\ Charles R Greathouse IV, Nov 07 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jun 19 2004
STATUS
approved