  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 Stirling
numbers of the first kind. For each $s$ we prove a central limit theorem and a
local limit theorem. This extends the de\,Moivre--Laplace central limit theorem
and Goncharov's result, that unsigned Stirling numbers of the first kind are
asymptotically normal. Herewith, we provide several applications.
