cp's OEIS Frontend

A287271 a(n) is the number of zeros of the Bernoulli B(n, x) polynomial in the open interval (-1, +1).

%I A287271 #27 Feb 16 2025 08:33:46
%S A287271 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 A287271 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 A287271 4,3,4,4,4,3,4,4,4,3,4,4,4,3,4,4,4,3,4
%N A287271 a(n) is the number of zeros of the Bernoulli B(n, x) polynomial in the open interval (-1, +1).
%C A287271 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 A287271 The first few Bernoulli polynomials are:
%C A287271 B(0,x) = 1
%C A287271 B(1,x) = x - 1/2
%C A287271 B(2,x) = x^2 - x + 1/6
%C A287271 B(3,x) = x^3 - 3/2 x^2 + 1/2 x
%C A287271 B(4,x) = x^4 - 2x^3 + x^2 - 1/30
%C A287271 Conjecture 1: for n > 32, a(n) = 3 for n odd and a(n) = 4 otherwise.
%C A287271 Conjecture 2: A284849(n) - a(n) > 0 if n = 6, 7,...,15 and n = 17.
%C A287271 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 A287271 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 A287271 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>
%F A287271 Conjectures from _Colin Barker_, Jan 22 2020: (Start)
%F A287271 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 A287271 a(n) = a(n-4) for n>16.
%F A287271 (End)
%e A287271 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 A287271 x1 = -0.2728865...-0.06497293...*i,
%e A287271 x2 = -0.2728865...+0.06497293...*i,
%e A287271 x3 =  0.2475407...,
%e A287271 x4 =  0.7524592...,
%e A287271 x5 =  1.272886...-.06497293...*i,
%e A287271 x6 =  1.272886...+.06497293...*i
%e A287271 with two roots x3 and x4 in the open interval (-1, +1).
%p A287271 f:= proc(n) sturm(sturmseq(bernoulli(n,x),x),x,-1,1) - `if`(n::odd and n > 1, 1, 0) end proc:
%p A287271 map(f, [$0..100]); # _Robert Israel_, May 29 2017
%t A287271 a[n_] := NSolve[-1 < x < 1 && BernoulliB[n, x] == 0, x, Reals, WorkingPrecision -> 100] // Length;
%t A287271 a /@ Range[0, 100] (* _Jean-François Alcover_, Oct 26 2020 *)
%Y A287271 Cf. A284849.
%K A287271 nonn
%O A287271 0,3
%A A287271 _Michel Lagneau_, May 22 2017
%E A287271 Corrected by _Robert Israel_, May 29 2017