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

0,3

Comments

The n-th Bernoulli polynomial is defined by the exponential generating function: t*exp(x*t)/(exp(t)-1) = Sum_{n>=0} bernoulli(n,x)/n!*t^n.

The first few Bernoulli polynomials are:

B(0,x) = 1

B(1,x) = x - 1/2

B(2,x) = x^2 - x + 1/6

B(3,x) = x^3 - 3/2 x^2 + 1/2 x

B(4,x) = x^4 - 2x^3 + x^2 - 1/30

Conjecture 1: for n > 32, a(n) = 3 for n odd and a(n) = 4 otherwise.

Conjecture 2: A284849(n) - a(n) > 0 if n = 6, 7,...,15 and n = 17.

Conjecture 1 is false. It appears that for n => 13, a(n) = 3 for n == 1 (mod 4) and a(n) = 4 otherwise. - Robert Israel, May 29 2017

Links

Table of n, a(n) for n=0..86.

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

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)

Example

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).

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

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 *)

Crossrefs

Cf. A284849.

Sequence in context: A349196 A112046 A076902 * A290884 A332383 A340327

Adjacent sequences: A287268 A287269 A287270 * A287272 A287273 A287274

Keyword

nonn

Author

Michel Lagneau, May 22 2017

Extensions

Corrected by Robert Israel, May 29 2017

Status

approved