A local limit theorem for the distribution of a part of the spectrum of a random binary function

We obtain a local limit theorem for the distribution of the vector (of growing dimension) consisting of some spectral coefficients of a random binary function of $n$ variables as $n\to\infty$. We correct a mistake in the asymptotic formula for the number of correlation-immune functions of order $k$ obtained in previous author's paper. We prove an asymptotic formula for the number of $(n,1,k)$-resilient functions as $n\to\infty$ and $k=k(n)=o(\sqrt n)$.