A284849 Number of zeros strictly inside the unit circle of the Bernoulli polynomial B(n,x).

0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 5, 6, 6, 5, 5, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3

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: for n > 63, a(n) = 3 for n odd and a(n) = 4 otherwise. - Charles R Greathouse IV, May 09 2017

Links

Table of n, a(n) for n=0..87.

Eric Weisstein's World of Mathematics, Bernoulli Polynomial

Formula

Conjectures from Colin Barker, Jan 22 2020: (Start)

G.f.: x*(1 + 2*x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 - x^12 - x^13 + x^14 - x^15 - x^16 - x^17 - x^18 - x^20 + x^22 - x^24 + x^26 - x^28 + x^30 - x^32) / ((1 - x)*(1 + x)).

a(n) = a(n-2) for n>33.

(End)

Example

a(6) = 4 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 four roots x1, x2, x3 and x4 in the unit circle.

Maple

for n from 0 to 90 do:it:=0:
y:=[fsolve(expand(bernoulli(n, x)), x, complex)]:for m from 1 to nops(y) do:if abs(y[m])<1 then it:=it+1:else fi:od: printf(`%d, `, it):od:

Prog

(PARI) a(n)=my(v=polroots(bernpol(n))); sum(i=1, #v, abs(v[i])<1) \\ Charles R Greathouse IV, May 09 2017

Crossrefs

Cf. A001898, A027641, A027642.

Sequence in context: A347578 A194163 A194253 * A274618 A176843 A238263

Adjacent sequences: A284846 A284847 A284848 * A284850 A284851 A284852

Keyword

nonn

Author

Michel Lagneau, May 09 2017

Status

approved