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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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified February 25 06:31 EST 2018. Contains 299643 sequences. (Running on oeis4.)