I want to know what is following sum coefficient looks like. We sum over all integers $p$, $q$ also we put the condition that $q$ is even. Also, it should depend on the parity of $k$ $$\bar{S}(k)=\sum_{p+q=k}[p]^{2m+1}q $$ The box symbol over $p$ denote that when p=0 it should be treated as $[0]=1$ I have seen the degree of the polynomial is 2m+3 in $k$. Can we claim that all the coefficient of the polynomial $S(k)$ is positive? For example when $m=0$ and $k$ is even we have $\bar{S}(k)=4 \binom{k/2+1}{3}+k$ hence all coefficient is positive.
Is it true in general all coefficient would be positive for $\bar{S} (k)?$
What about the sum below where r is even depending on the parity is all the coefficient is non nengative or it has general closed formula in $k$? $$\bar{R}(k)=\sum_{p+q+r=k}[p]^{2m+1}[q]^{2m'+1}r.$$