Evaluating a Combinatorial summation $$aC_0bC_0 + 2aC_1bC_1 + 3 aC_2bC_2 + 4 aC_3bC_3+\cdots+a\cdot aC_abC_a$$ How to go about evaluating this? I know the basic result
$$aC_0bC_a + aC_1bC_{a-1} +  aC_2bC_{a-2} +  aC_3bC_{a-3}+\cdots+a\cdot aC_abC_0 = (a+b)C_a$$

From the comments you already have the form $$\sum_{k=0}^a(k+1)\binom{a}k\binom{b}{b-k}\;.$$ Split it in two to get
$$\sum_{k=0}^a(k+1)\binom{a}k\binom{b}{b-k}=\sum_{k=0}^ak\binom{a}k\binom{b}{b-k}+\sum_{k=0}^a\binom{a}k\binom{b}{b-k}\;,$$
apply the identity
$$k\binom{a}k=a\binom{a-1}{k-1}\;,$$
and apply the Vandermonde identity to each summation.
