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You can rewrite it in different notation with sum limits if you want. You just need to use a different way to express the constraint on which terms to include in the sum. E.g. with the [Iverson bracket](https://en.wikipedia.org/wiki/Iverson_bracket) notation the LHS becomes

$$\sum_{k_1 = 0}^m \sum_{k_2 = 0}^m \cdots \sum_{k_p = 0}^m [k_1+\cdots +k_p = m] {n_1\choose k_1} {n_2\choose k_2} \cdots {n_p\choose k_p}$$