Yeah it is, but i thought:
z1z2 + z1z3 + z1z4 + z1z5 + ... + z1zn +
z2z3 + z2z4 + z2z5 + ... + z2zn +
all the way
+ z(n-1)zn = 0
but then, your equation only involves with z1z2, z2z3, z3z4, z4z5.. and does not involve z1z3 (number of the root that is not next to each other)..