

A287271


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


0



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
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OFFSET

0,3


COMMENTS

The nth 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 zetafunction and Bernoulli polynomials, Journal of Mathematical Analysis and Applications 305 (2005), 712721.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial


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


CROSSREFS

Cf. A284849.
Sequence in context: A059906 A112046 A076902 * A290884 A049113 A055093
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



