A287271 - OEIS

%I #26 Oct 26 2020 08:05:53

%S 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,

%T 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,

%U 4,3,4,4,4,3,4,4,4,3,4,4,4,3,4,4,4,3,4

%N a(n) is the number of zeros of the Bernoulli B(n, x) polynomial in the open interval (-1, +1).

%C 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.

%C The first few Bernoulli polynomials are:

%C B(0,x) = 1

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

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

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

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

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

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

%C 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

%H A. P. Veselov and J. P. Ward, <a href="https://doi.org/10.1016/j.jmaa.2004.12.046">On the real zeros of the Hurwitz zeta-function and Bernoulli polynomials</a>, Journal of Mathematical Analysis and Applications 305 (2005), 712-721.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>

%F Conjectures from _Colin Barker_, Jan 22 2020: (Start)

%F 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)).

%F a(n) = a(n-4) for n>16.

%F (End)

%e 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:

%e x1 = -0.2728865...-0.06497293...*i,

%e x2 = -0.2728865...+0.06497293...*i,

%e x3 =  0.2475407...,

%e x4 =  0.7524592...,

%e x5 =  1.272886...-.06497293...*i,

%e x6 =  1.272886...+.06497293...*i

%e with two roots x3 and x4 in the open interval (-1, +1).

%p f:= proc(n) sturm(sturmseq(bernoulli(n,x),x),x,-1,1) - `if`(n::odd and n > 1, 1, 0) end proc:

%p map(f, [$0..100]); # _Robert Israel_, May 29 2017

%t a[n_] := NSolve[-1 < x < 1 && BernoulliB[n, x] == 0, x, Reals, WorkingPrecision -> 100] // Length;

%t a /@ Range[0, 100] (* _Jean-François Alcover_, Oct 26 2020 *)

%Y Cf. A284849.

%K nonn

%O 0,3

%A _Michel Lagneau_, May 22 2017

%E Corrected by _Robert Israel_, May 29 2017