elementary number theory - $(1 cdot 2 cdot ... cdot m)^w+ (2 cdot 3 cdot ... cdot (m+1))^w+...+(n cdot(n+1)+ cdot ... cdot (n+m-1))^w=?$

I started to read some book on elementary number theory and preliminary chapter asks to establish some formulas by mathematical induction:

There is this formula:

$$1+2+...+n= \dfrac {n(n+1)}{2}$$

And this one:

$$1\cdot 2 + 2\cdot 3 + ... + n\cdot (n+1)= \dfrac {n(n+1)(n+2)}{3}$$

With some pencil-and-paper work I established that this should hold:

$$1\cdot 2 \cdot ... \cdot m + 2\cdot 3 \cdot ... \cdot (m+1) + ... + n \cdot (n+1) \cdot ... \cdot (n+m-1)= \dfrac {n(n+1)...(n+m-1)(n+m)}{m+1}$$

I did not prove this formula that I established but just checked some cases and it seems to hold.

It can be written in the form of a hockey-stick identity, I think, so it holds.

Now, I know about generalization of first formula that goes like this (Faulhaber´s formula):

$$1^w + 2^w + ... + n^w=\dfrac {1}{w+1} \cdot \displaystyle \sum_{j=0}^{w} { {w+1} \choose j} B_j n^{w+1-j}$$

How do the generalization $$(1 \cdot 2 \cdot ... \cdot m)^w+ (2 \cdot 3 \cdot ... \cdot (m+1))^w+...+(n \cdot(n+1)+ \cdot ... \cdot (n+m-1))^w=?$$ look like? That is, what is on the right side?