A287271 a(n) is the number of zeros of the Bernoulli B(n, x) polynomial in the open interval (-1, +1).
0, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4
Offset: 0
Keywords
Examples
a(6) = 2 because the zeros of B(6,x) = x^6 - 3x^5 + 5/2 x^4 - 1/2 x^2 + 1/42 are: x1 = -0.2728865...-0.06497293...*i, x2 = -0.2728865...+0.06497293...*i, x3 = 0.2475407..., x4 = 0.7524592..., x5 = 1.272886...-.06497293...*i, x6 = 1.272886...+.06497293...*i with two roots x3 and x4 in the open interval (-1, +1).
Links
- A. P. Veselov and J. P. Ward, On the real zeros of the Hurwitz zeta-function and Bernoulli polynomials, Journal of Mathematical Analysis and Applications 305 (2005), 712-721.
- Eric Weisstein's World of Mathematics, Bernoulli Polynomial
Crossrefs
Cf. A284849.
Programs
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Maple
f:= proc(n) sturm(sturmseq(bernoulli(n,x),x),x,-1,1) - `if`(n::odd and n > 1, 1, 0) end proc: map(f, [$0..100]); # Robert Israel, May 29 2017
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Mathematica
a[n_] := NSolve[-1 < x < 1 && BernoulliB[n, x] == 0, x, Reals, WorkingPrecision -> 100] // Length; a /@ Range[0, 100] (* Jean-François Alcover, Oct 26 2020 *)
Formula
Conjectures from Colin Barker, Jan 22 2020: (Start)
G.f.: x*(1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 2*x^9 + 2*x^10 + 2*x^15 - x^19) / ((1 - x)*(1 + x)*(1 + x^2)).
a(n) = a(n-4) for n>16.
(End)
Extensions
Corrected by Robert Israel, May 29 2017
Comments