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

%I A284849 #25 Feb 16 2025 08:33:43
%S A284849 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,
%T A284849 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,
%U A284849 4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3,4,3
%N A284849 Number of zeros strictly inside the unit circle of the Bernoulli polynomial B(n,x).
%C A284849 The n-th Bernoulli polynomial is defined by the exponential generating function:
%C A284849 t*exp(x*t)/(exp(t)-1) = Sum_{n>=0} bernoulli(n,x)/n!*t^n.
%C A284849 The first few Bernoulli polynomials are:
%C A284849 B(0,x) = 1
%C A284849 B(1,x) = x - 1/2
%C A284849 B(2,x) = x^2 - x + 1/6
%C A284849 B(3,x) = x^3 - 3/2 x^2 + 1/2 x
%C A284849 B(4,x)= x^4 - 2x^3 + x^2 - 1/30
%C A284849 Conjecture: for n > 63, a(n) = 3 for n odd and a(n) = 4 otherwise. - _Charles R Greathouse IV_, May 09 2017
%H A284849 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>
%F A284849 Conjectures from _Colin Barker_, Jan 22 2020: (Start)
%F A284849 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)).
%F A284849 a(n) = a(n-2) for n>33.
%F A284849 (End)
%e A284849 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:
%e A284849 x1 = -0.2728865...-0.06497293...*i,
%e A284849 x2 = -0.2728865...+0.06497293...*i,
%e A284849 x3 =  0.2475407...,
%e A284849 x4 =  0.7524592...,
%e A284849 x5 =  1.272886...-.06497293...*i,
%e A284849 x6 =  1.272886...+.06497293...*i
%e A284849 with four roots x1, x2, x3 and x4 in the unit circle.
%p A284849 for n from 0 to 90 do:it:=0:
%p A284849 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:
%o A284849 (PARI) a(n)=my(v=polroots(bernpol(n))); sum(i=1,#v,abs(v[i])<1) \\ _Charles R Greathouse IV_, May 09 2017
%Y A284849 Cf. A001898, A027641, A027642.
%K A284849 nonn
%O A284849 0,3
%A A284849 _Michel Lagneau_, May 09 2017