Hockey-Stick Theorem for Multinomial Coefficients

Question

Pascal's triangle has this famous hockey stick identity. $$ \binom{n+k+1}{k}=\sum_{j=0}^k \binom{n+j}{j}$$ Wonder what would be the form for multinomial coefficients?

Answer 1

$$\binom{a_1+a_2+\cdots+a_t}{a_1,a_2,\cdots,a_t}=\sum_{i=2}^t \sum_{j=1}^{i-1} \sum_{k=1}^{a_i} \binom{ a_1+a_2+\cdots+a_{i-1}+k }{a_1,a_2,\cdots,a_j-1,\cdots,a_{i-1},k }$$