Hamming weights of symmetric Boolean functions

It has been known since 1996 (work of Cai et al.) that the sequence of Hamming weights {wt(f_n(x₁,…,x_n))}, where f_n is a symmetric Boolean function of degree d in n variables, satisfies a linear recurrence with integer coefficients. In 2011, Castro and Medina used this result to show that a 2008 conjecture of Cusick, Li and Stănică about when an elementary symmetric Boolean function can be balanced is true for all sufficiently large n. Quite a few papers have been written about this conjecture, but it is still not completely settled. Recently, Guo, Gao and Zhao proved that the conjecture is equivalent to the statement that all balanced elementary symmetric Boolean functions are trivially balanced. This motivates the further study of the weights of symmetric Boolean functions f_n. In this paper we prove various new results on the trivially balanced functions. We also determine a period (sometimes the minimal one) for the sequence of weights wt(f_n) modulo any odd prime p, where f_n is any symmetric function, and we prove some related results about the balanced symmetric functions.