From the multinomial formula we have

$$(x_1 + x_2 + \dotsb + x_m)^n = \sum_{k_1+k_2+\dotsb+k_m=n, \ k_1, k_2, \dotsc, k_m \geq 0} {n \choose k_1, k_2, \dotsc, k_m} \prod_{t=1}^m x_t^{k_t}\,.$$

I wish to evaluate the RHS, but with the additional constraint that each exponent $k_i$ can not exceed some positive integer $a<n$. That is, $$ \sum_{k_1+k_2+\dotsb+k_m=n, \ 0\leq k_1, k_2, \dotsc, k_m \leq a} {n \choose k_1, k_2, \dotsc, k_m} \prod_{t=1}^m x_t^{k_t}\,. $$ While I can obviously calculate this, my question is whether there is a simple way to express the polynomial with the restriction on the exponents. This is to ultimately facilitate computation. For example, in the case where $a=n-1$ we could write this as $$ \left(\sum_{i=1}^m x_i\right)^n - \sum_{i=1}^m x_i^n. $$ Is there a simple way of expressing the polynomial for other integer values $0< a<n$?