Abstract. Let $F_{p,t} (n)$ denote the number of the coefficients of $(x_1 + x_2 +\cdots + x_t)^j$, $0\le j\le n-1$, which are not divisible by the prime $p$. Then we have $\alpha(p,t) = \limsup F_{p,t} (n)/n^\theta =1$, and $\beta(p,t)=\liminf F_{p,t} (n) /n^\theta $ can be calculated to a given precision, where $\theta = \log{p+t-1\choose t} \big/ \log p$.

AMS Subject Classification (1991): 11A63, 11K16

Received October 28, 1997 and in revised form January 26, 1998. (Registered under 2643/2009.)