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

0,3

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.

Links

Table of n, a(n) for n=0..69.

Example

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.

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);

Crossrefs

Cf. A007318, A214292.

Sequence in context: A269757 A308973 A309279 * A245352 A099033 A187786

Adjacent sequences: A382016 A382017 A382018 * A382020 A382021 A382022

Keyword

nonn

Author

Michel Lagneau, Mar 12 2025

Status

approved