cp's OEIS Frontend

A382019 Number of zeros (counted with multiplicity) inside and on the unit circle of the polynomial P(n,z) = Sum_{k=0..n} T(n,k)*z^k where T(n,k) = A214292(n,k) is the first differences of rows in Pascal's triangle.

%I A382019 #26 Mar 25 2025 14:02:23
%S A382019 0,1,2,3,4,5,6,5,6,7,8,9,10,9,10,11,12,13,14,13,14,15,16,17,18,17,18,
%T A382019 19,20,21,22,21,22,23,24,25,26,25,26,27,28,29,30,29,30,31,32,33,34,33,
%U A382019 34,35,36,37,38,37,38,39,40,41,42,41,42,43,44,45,46,45,46,47
%N A382019 Number of zeros (counted with multiplicity) inside and on the unit circle of the polynomial P(n,z) = Sum_{k=0..n} T(n,k)*z^k where T(n,k) = A214292(n,k) is the first differences of rows in Pascal's triangle.
%C A382019 The polynomial is P(n,z) = z^(n+1) - ((z-1)*(z+1)^(n+1) +1)/z.
%C A382019 A root z (real or complex) is in or on the unit circle when its magnitude abs(z) <= 1.
%e A382019 a(4)=4 because P(4,z)= 4 + 5*z -5*z^3 -4*z^4  with 4 roots z1, z2, z2, z4 on the unit circle : z1 = -1, z2 = +1, z3 = -.625000 -.7806247*i, z4 = -.625000 +.7806247*i.
%e A382019 a(6)=6 because P(6,z)= 6 + 14*z +14*z^2 -14*z^4-14*z^5-6z^6  with 6 roots on the unit circle:
%e A382019   z1 = -1,
%e A382019   z2 = +1,
%e A382019   z3 = -.6666666667 - .7453559925*i,
%e A382019   z4 = -.6666666667 + .7453559925*i,
%e A382019   z5 = -.500000000 - .8660254038*i,
%e A382019   z6 = -.500000000 + .8660254038*i.
%p A382019 A382019:=proc(n) local m,y,it:
%p A382019 y:=[fsolve(add((binomial(n+1,k+1)-binomial(n+1,k))*x^k,k=0..n),x,complex)]:it:=0:
%p A382019  for m from 1 to nops(y) do:
%p A382019           if ((Re(y[m]))^2+(Im(y[m]))^2)<=1
%p A382019           then
%p A382019          it:=it+1:else fi:
%p A382019    od: A382019(n):=it:end proc:
%p A382019 seq(A382019(n),n=1..70);
%Y A382019 Cf. A007318, A214292.
%K A382019 nonn
%O A382019 0,3
%A A382019 _Michel Lagneau_, Mar 12 2025