Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$ with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=0}^t HF(T/I,i)=\prod_{i=2}^n a_i.$$ I know that $I$ is a complete intersection ideal, so $T/I$ is Artinian and Gorenstein, so there exists $\tau$ s.t. $\dim_k [T/I]_{\tau}=1$ (and of course I can cut this sum after the $\tau$ index), but I can't see why I can pass from this summatory to a product of esponents. T tried some combinatorial proofs without much success.

This question is taken by these two paper (resp. page 8 and 2, both at the bottom), which I link you:

https://www.sciencedirect.com/science/article/pii/S0021869312003730

https://arxiv.org/abs/1110.0745

Can anyone help me? Thanks in advance.