

A094937


Number of real roots of the nth Bernoulli polynomial B(n,x).


0



0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 12, 13, 14, 15, 16, 17, 14, 15, 16, 17, 18, 15, 16
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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. 754775.
A. P. Veselov and J. P. Ward, On the real zeros of the Hurwitz zetafunction and Bernoulli polynomials. J. Math. Anal. Appl. 305 (2005), no. 2, 712721.


LINKS

Table of n, a(n) for n=0..60.
A. P. Veselov and J. P. Ward, On the real roots of the Bernoulli polynomials and the Hurwitz zetafunction, 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}] (* JeanFrançois Alcover, Sep 13 2012 *)


PROG

(PARI) a(n)=polsturm(sum(i=0, n, binomial(n, i)*bernfrac(i)*x^(ni)))
(PARI) a(n)=my(e=1e29, v=polroots(bernpol(n))); sum(i=1, #v, abs(imag(v[i])) <= abs(v[i])*e) \\ Charles R Greathouse IV, Nov 07 2012


CROSSREFS

Sequence in context: A328943 A173525 A070772 * A215089 A329243 A161768
Adjacent sequences: A094934 A094935 A094936 * A094938 A094939 A094940


KEYWORD

nonn


AUTHOR

Benoit Cloitre, Jun 19 2004


STATUS

approved



