What is the asymptotic growth of the terms in this expression?

Question

Let's say we have $k$ variables in this expression: $(n_1+n_2+n_3+...+n_k)^2$. When you expand it, you get something like $n_1^2+n_1n_2+n_2^2+...$ , using the multinomial theorem. Now, if we multiply each term of the expansion out like this: $(m_1n_1^2+c_1)(m_2n_1n_2+c_2)(m_3n_2^2+c_3)...$ and expand it, is it true that the number of terms is $O(2^{k^2})$?

Answer 1

No. The expansion of $A=(\sum_{i=1}^kn_i)^2$ has $k(k+1)/2$ terms in it. The expansion of $\prod_{i=1}^{k(k+1)/2}(T_i+c_i)$, where $T_i$ is the $i$ term in the expansion of $A$, has $2^{k(k+1)/2}$ terms.