Analytic and combinatorial approaches to a weighted Catalan sum

We analyze a weighted convolution of Catalan numbers $$\sum_{k=0}^{n} \binom{2k}{k}\binom{2(n-k)}{n-k} a^k = \sum_{k=0}^{n} (k+1)(n-k+1) C_k C_{n-k} a^k,$$ emphasizing its combinatorial, analytic, and probabilistic aspects. We derive a compact closed form in terms of the Gauss hypergeometric function ${}_2F_1(-n,1/2;1;1-a)$, valid for all complex values of the parameter $a$. The sum admits a natural interpretation in terms of return probabilities of independent simple random walks, linking weighted convolutions of central binomial coefficients to classical probability theory. Furthermore, a refinement via Narayana numbers highlights the contribution of peak distributions in pairs of Dyck paths, providing a finer combinatorial perspective. An integral representation is also proposed, suggesting a connection with orthogonal polynomials and spectral measures. Our approach illustrates how analytic and probabilistic techniques complement combinatorial reasoning in evaluating complex sums.