what's the summation of this finite sequence?

Question

$a$ and $b$ are positive integers. The summation is $$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}} {a + b - x}\\ b \end{array}} \right)} .$$ Any closed-form expression?

I thought it should have. And maybe there is some physical meaning behind it.

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Sorry, I've simplified the problem, and now it becomes easier.

Answer 1

This is the binomial identity $\sum_{m=0}^n\binom{m}{j}\binom{n-m}{k-j} = \binom{n+1}{k+1}$ with $j = 1$, $n = a+ b$ and $k = b+1$.

Answer 2

Wolfram Alpha produces:$$\sum_{x=1}^a x \binom{a+b-x}{b} =\frac{(a+b) (a+b+1) \binom{a+b-1}{b}}{(b+1) (b+2)}$$ Full simplification of RHS: $$\frac{\Gamma (a+b+2)}{\Gamma (a) \Gamma (b+3)}$$

Answer 3

Some experimentation gives $\dbinom{a+b+1}{b+2}$. This is the correct answer for $1\le a, b\le 5$.