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Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts

For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function. Here $(x)_n$ is the Pochhammer symbol and ${a\...