A382108 Number of zeros (counted with multiplicity) 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, 3, 4, 3, 6, 5, 6, 5, 6, 7, 8, 9, 10, 3, 8, 7, 10, 9, 10, 7, 10, 11, 8, 11, 12, 9, 10, 11, 14, 11, 14, 11, 12, 13, 12, 13, 12, 15, 12, 7, 18, 19, 16, 11, 14, 11, 14, 11, 18, 11, 18, 15, 18, 19, 22, 7, 16, 21, 20, 17, 22, 15, 18, 21, 20, 25, 20

Offset

0,3

Links

Table of n, a(n) for n=0..72.

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:
x1 = -1
x2 = +1
x2 = -.6666666667 - .7453559925*i
x3 = -.6666666667 + .7453559925*i
x5 = -.500000000 - .8660254038*i
x6 = -.500000000 + .8660254038*i

Maple

A382108:=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:
    A382108(n):=it:end proc:seq(A382108(n), n=1..80);

Crossrefs

Cf. A007318, A214292, A382019 (on and inside the circle).

Sequence in context: A352425 A104148 A286450 * A245344 A323074 A195153

Adjacent sequences: A382105 A382106 A382107 * A382109 A382110 A382111

Keyword

nonn

Author

Michel Lagneau, Mar 15 2025

Status

approved