I am currently stuck with a formula I am trying to evaluate. At one point there is the following sum $$ \sum_{\substack{ (l_1,\ldots,l_m) \in \{ 1,\ldots,l \}^m \\ l_1 + \cdots + l_m = l}} \prod_{r=1}^m \binom{k_r}{l_r}, \quad k_r \in \mathbb{R}, m, l \in \mathbb{N}, $$ and I have no idea how I could possibly simplify this sum. Note that the numbers $l_1,\ldots,l_m$ have to be positive. Also, to be more specific about the values of $k_r$, it's either a natural number or a positive half integer. Could anyone help me with this problem? I would be immensely grateful for any help.
2026-04-06 09:06:09.1775466369
Certain sum of products of binomial coefficients with constraints
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