Given two non-negative integers $m,n$. Define $F_{m,n}$ as the set of all non negative integer sequence $(i_0,i_1,\ldots,i_n)$ which satisfiy
$$i_1+2i_2+\cdots+(n-1)i_{n-1} = m$$
and
$$i_0+i_1+\cdots+i_n = n.$$
Then how to estimate
$$f_{m,n} = \sum_{(i_0,i_1,\ldots,i_n)\in F_{m,n}} \binom{n}{i_0,i_1,\ldots,i_n} 2^{m-f(A)} $$
where
$$f(A) = \sum_{t=0}^{n-1} \delta_0^{i_t}$$
and $\delta_i^j$ is the Kronecker notation. Moreover, how to estimate
$$P = \frac{1}{2^{n^2}} \sum_{m=0}^{n(n-1)} f_{m,n}.$$