Refined asymptotic estimates for the number of $(n,m,k)$-resilient Boolean mappings

For linear combinations of coordinate functions of a random Boolean mapping, a local limit theorem for the distribution of subsets of their spectral coefficients is improved. By means of this theorem, we obtain an asymptotic formula for the $R(m,n,k)|$ –the number of $(n,m,k)$-resilient functions as $n\to\infty$, $m\in\{1,2,3,4\}$ and $k\leq\frac{n(1-\varepsilon)}{5+2\log _2n}$ for any $0<\varepsilon <1$, $k=\mathrm O(\frac n{\ln n})$:
\begin{gather*} \log _2|R(m,n,k)|\sim m2^n-(2^m-1)\left(\frac{n-k}2{n\choose k}+\log _2\sqrt\frac\pi2\sum_{s=0}^k{n\choose s}\right)+\\ +(2\cdot3^{m-2}-1)\mathrm{Ind}\{m\neq1\}\sum_{s=0}^k{n\choose s}. \end{gather*}
Also, we obtain upper and lower asymptotic estimates for the number $|R(m,n,k)|$ as $n\to\infty$, $k(5+2\log _2n)+5m\le n(1-\varepsilon)$ for any $0<\varepsilon<1$:
\begin{gather*} -\varepsilon_1(m-1)\sum_{s=0}^k{n\choose s}<\log _2|R(m,n,k)|-m2^n+(2^m-1)\left(\frac{n-k}2{n\choose k}+\log_2\sqrt\frac\pi2\sum_{s=0}^k{n\choose s}\right)<\\ <\varepsilon_2(m-2)(2^m-1)\sum_{s=0}^k{n\choose s}+\sum_{s=0}^k{n\choose s}\qquad\text{for any}\quad\varepsilon_1,\varepsilon_2\quad(0<\varepsilon_1,\varepsilon_2<1). \end{gather*}