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

Author

Michel Lagneau, May 09 2017

Keywords

• nonn

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

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Bernoulli Polynomial

Crossrefs

Cf. A001898, A027641, A027642.

Programs

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:

PARI

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

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)