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A094937 Number of real roots of the n-th 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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A119281 A173525 A070772 * A215089 A161768 A213925

Adjacent sequences:  A094934 A094935 A094936 * A094938 A094939 A094940

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Jun 19 2004

STATUS

approved

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Last modified March 20 07:42 EDT 2019. Contains 321345 sequences. (Running on oeis4.)