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A094937
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Number of real roots of the n-th Bernoulli polynomial B(n,x).
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0
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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
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0,3
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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a(n) = 2n/(Pi*e) + O(log n).
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MATHEMATICA
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a[n_] := CountRoots[ BernoulliB[n, x], x]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 13 2012 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A328943 A173525 A070772 * A215089 A329243 A161768
Adjacent sequences: A094934 A094935 A094936 * A094938 A094939 A094940
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre, Jun 19 2004
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STATUS
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approved
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