The maximum number of ones and zeros and their distribution among the coefficients of some polynomials is given. It is shown that the coefficients of the polynomials $g_1 ( x ) = ( x^n + 1)/p( x )$ and $g_2 ( x ) = ( x^n + 1)[ ( x + 1)p( x )]$ and other relatedpolynomials have special distribution when $p( x )$ is a primitive polynomial, and these coefficients contain the maximum possible number of ones and zeros.