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Question: How to prove the following conjecture? ...

Conjecture: For all integers $r\ge 0$ and $b\ge 3$,

$$ {\LARGE\sum}_{j=2}^\infty{\LARGE\sum}_{i=j-2}^\infty{(r)_i(b)_j\over (r+b)_{i+j}}\binom{i}{j-2}(i+j)\ =\ \left(1+{r\over b-1}\right)\left(2+{r\over b-2}\right)$$

where the Pochhammer symbols are defined as $$(a)_n = \begin{cases} 1 &\text{ if }n = 0 \\ a(a+1)(a+2)\cdots (a+n-1) &\text{ if } n > 0. \end{cases}$$

Note: This is motivated by a previous as-yet-unanswered question. The LHS is the expected value of the first-occurrence time of $\tt BB$ in the sequence of colors drawn in a Polya Urn Process that initially has $b$ $\tt{B}$lue balls and $r$ $\tt{R}$ed balls. The RHS is the result of inspecting pseudorandom simulations of the process for a variety of $(b,r)$ values.

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