The Walsh transform f̂ of a quadratic function f: F(pn) → Fpsatisfies |f̂| ∈ {0,pn+s/2} for an integer 0 ≤ s ≤ n-1, depending on f. In this paper, quadratic functions of the form Fp,n(x) = Trn(Σi=0kaixpi+1) are studied, with the restriction that ai∈ Fp, 0 ≤ i ≤ k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2 m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.