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A284849
Number of zeros strictly inside the unit circle of the Bernoulli polynomial B(n,x).
1
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, May 09 2017
STATUS
approved