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 $k$? 
$$\bar{R}(k)=\sum\_{p+q+r=k}[p]^{2m+1}[q]^{2m'+1}r.$$



If the coefficient of the polynomial positive
When $m=1$ and $k$ is odd, I compute that $\bar{S}(k) = (3k^5-20k^3+17k)/120$, which has a negative coefficient.


