How to use generating functions to partially sum multiple integer sequences?

Question

Let's say I want to find the following double sum $$ \sum_{k=1}^mk\sum_{n=1}^kn={1\over24}m(1+m)(2+m)(1+3m) $$ but using a generating function for the involved sums. The polynomial generating function for $\sum_{n=1}^kn$ is simply $$ F(x)=\sum_{n=1}^knx^n={x+x^{1+k}(kx-k-1)\over (x-1)^2} $$ and similarly for the other sum (presumably with a different variable, say $y$). But how do I combine the two generating functions to be able to take a limit $x,y\to1$ to get the partial sum? It does not look like a convolution to me.

Answer 1

Actually it is 'easy'. The second generating function is simply $$ G(x,y)=\sum_{k=1}^mkF(x)y^k=\sum_{k=1}^mk\frac{x+x^{k+1} (k x-k-1) }{(x-1)^2}y^k $$ yielding a complicated expression I cannot paste. We get the correct result upon taking the limit $x,y\to1$.