Stirling's Original Asymptotic Series from a Formula like one of Binet's and its Evaluation by Sequence Acceleration

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of $\ln z!$ for large $z$. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the sequence acceleration method known as Levin's $u$-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.