A094937 - OEIS

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A094937

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

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. 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

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 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

Sequence in context: A328943 A173525 A070772 * A215089 A329243 A375636