  The series $$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m
\{array})}$$ is evaluated in non-recursive closed and analytically continued
beyond its domain of convergence $0\le |z|<4$ for $k=0,1,2,\...$. From this we
provide a firm basis for Lehmer's observation that $\pi$ emerges from the
limiting behavior of $S_k(2)$ as $k\rightarrow\infty$.
