Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\lambda$.

I want to know if following results appear in literature if anyone can give me a reference or proof $$ \sum_{\lambda\in \text{different hook of size d}} \frac{1}{|\lambda|!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-c(\Box)h}$$ that the above equation equal to \begin{equation} \frac{(2d-2)!}{ (d-1)!}h^{d-1}\prod_{i=1}^{d-1} \frac{1}{(1+ih)(1-ih)} \end{equation}

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