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The sum of values of the Hilbert function is equal to the vector space dimension of the algebra: $$ \sum_{i=0}^{\infty} HF(T/I,i) = \dim(T/I). $$ In this case, the vector space $T/I$ has a basis consisting of monomials $x_1^{b_1} \dotsm x_n^{b_n}$ with $0 \leq b_i \leq a_i-1$ for each $i$ (where I am taking $a_1=1$). So the dimension is $\prod_{i=1}^n a_i$. Or, this follows from general facts about complete intersections.