Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum of all terms in the sequence, I want to find the generating series of the tuple $(S_k,w)$.
For example, $s = 3,2,2,2,1,0,0,0$ is a member of $S_k$, and $w(s) = 10$.
I attempted by noticing that the coefficient of $x^i$ is basically finding the number of ways to represent $i$ as sum of $k$ numbers, order doesn't matter and $0$ allowed. This is basically adding the number of integer partition of $i$ with 1 part, with the number of integer partition of $i$ with 2 parts, etc. all the way to k parts. The reason we have to do this and not just take the integer partition is because $0$ is allowed as a term of the sequence, so say for $i=6,k=4$, then $6,0,0,0$ is a valid sequence with length $4$ and sums to $6$, but it is integer partition into only 1 part.
Based on this idea, the generating series would be $\sum_{i=0}^{\infty}{(\sum_{j=1}^k{p_j(i)}) x^i}$, where $\sum_{j=1}^k{p_j(i)}$ is the number of partitions for $i$ with $j$ parts. This is how far I can get, but I'm wondering if I can find a closed form for the expression, or at least a cleaner and simpler way to represent this.