I have checked the following identity (1) below for $n\leq 40$ with a computer. Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let $Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let $D_n=x^2Z_{n-1}Z_{n-2}-2nxZ_nZ_{n-2}+Z_nZ_{n-1}$. Then, I conjecture that
$$ D_n=\sum_{0 \leq i \leq j \leq n-2} (n-j)!(n-j-1)!(j-i)! \binom{n-2}{j}\binom{j}{i}\binom{2n+1-i-j}{j-i}x^{i+j}\tag{1} $$
I tried my usual tools (induction, WS method, using similar identities) and failed. Any help appreciated.
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