The main purpose of the paper is to obtain a single formula for calculating feeding coefficients for optimum beam patterns. The new formula derived is equivalent to a set of formulae given by DuHamel, but is much simpler in form and more convenient for computation. It is applicable to Riblet's generalization of Dolph's procedure for finding optimum beam patterns in the design of linear arrays, for an odd number of elements when the distance between sources is less than half the wavelength. Whereas DuHamel's formulae involve a number of separate summation expressions, each of which has terms involving considerable multiplication and division, the corresponding terms in the present, more concise, expression have simpler integral coefficients involving the product of only two binomial coefficients. The mathematical problem solved is to find the explicit formula forbmin the expression of Tn(ax+b) in the form Σm=0nbmTm(x), where Tm(x) = cos(marc cosx) for m>0, T0(x) = ½.