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Are there any identities involving this product of Harmonic Numbers?

$$u_{n,m}=\prod_{k=1}^{n-1} H_k \prod_{k=1}^m H_{k+n}$$

My motivation is an attempt to study a series of the form $$\sum_{n=1}^\infty \frac{u_{n,m}}{n^s}x^n$$

Perhaps either the product or sum-of-products has an integral form? Here is an example of an integral form for a sum of a product of two harmonic numbers.

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