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On the divisibility by 2 of the Stirling numbers of the second kind. (English) Zbl 0808.11017
Summary: We characterize the divisibility by 2 of the Stirling numbers of the second kind, $$S(n,k)$$, where $$n$$ is a sufficiently high power of 2. Let $$\nu_ 2(r)$$ denote the highest power of 2 that divides $$r$$. We show that there exists a function $$L(k)$$ such that, for all $$n\geq L(k)$$, $$\nu_ 2 (k! S(2^ n, k))= k-1$$ hold, independently from $$n$$. (The independence follows from the periodicity of the Stirling numbers modulo any prime power.) For $$k\geq 5$$, the function $$L(k)$$ can be chosen so that $$L(k)\leq k-2$$. We determine $$\nu_ 2 (k! S(2^ n+u, k))$$ for $$k>u\geq 1$$, in particular for $$u=1$$, 2, 3, and 4. We show how to calculate it for negative values, in particular for $$u=-1$$. The characterization is generalized for $$\nu_ 2 (k! S(c\cdot 2^ n+ u,k))$$, where $$c>0$$ denotes an arbitrary odd integer.

##### MSC:
 11B73 Bell and Stirling numbers 11B50 Sequences (mod $$m$$)