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.

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, 19, 20, 21, 22, 21, 22, 23, 24, 25, 26, 25, 26, 27, 28, 29, 30, 29, 30, 31, 32, 33, 34, 33, 34, 35, 36, 37, 38, 37, 38, 39, 40, 41, 42, 41, 42, 43, 44, 45, 46, 45, 46, 47

Offset: 0

Author

Michel Lagneau, Mar 12 2025

Keywords

• nonn

Comments

The polynomial is P(n,z) = z^(n+1) - ((z-1)*(z+1)^(n+1) +1)/z.

A root z (real or complex) is in or on the unit circle when its magnitude abs(z) <= 1.

Examples

			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.
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:
  z1 = -1,
  z2 = +1,
  z3 = -.6666666667 - .7453559925*i,
  z4 = -.6666666667 + .7453559925*i,
  z5 = -.500000000 - .8660254038*i,
  z6 = -.500000000 + .8660254038*i.
		

Crossrefs

Cf. A007318, A214292.

Programs

Maple

A382019:=proc(n) local m,y,it:
y:=[fsolve(add((binomial(n+1,k+1)-binomial(n+1,k))*x^k,k=0..n),x,complex)]:it:=0:
 for m from 1 to nops(y) do:
          if ((Re(y[m]))^2+(Im(y[m]))^2)<=1
          then
         it:=it+1:else fi:
   od: A382019(n):=it:end proc:
seq(A382019(n),n=1..70);