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We construct a new parametrization of double sequences $\{A_{n,k}(s)\}_{n,k}$between $A_{n,k}(0)= \binom{n-1}{k-1}$ and $A_{n,k}(1)=\frac{1}{n!}\stirl{n}{k}$, where $\stirl{n}{k}$ are the unsigned Stirlingnumbers of the first kind. For each $s$ we prove a central limit theorem and alocal limit theorem. This extends the de\,Moivre--Laplace central limit theoremand Goncharov's result, that unsigned Stirling numbers of the first kind areasymptotically normal. Herewith, we provide several applications.

Variations of Central Limit Theorems and Stirling numbers of the First Kind