On the Divisibility by 2 of the Stirling Numbers of the Second Kind
نویسنده
چکیده
In this paper we characterize the divisibility by 2 of the Stirling number of the second kind, S(n; k); where n is a su ciently high power of 2. Let 2(r) denote the highest power of 2 which divides r. We show that there exists a function L(k) such that for all n L(k); 2 k!S(2; k) = k 1 hold, independently from n: (Here the independence follows from the periodicity of the Stirling numbers modulo any prime power.) For k 5, the function L(k) can be chosen so that L(k) k 2: We determine 2 k!S(2 + u; k) for k > u 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 2 k!S(c 2 + u; k) where c > 0 denotes an arbitrary odd integer.
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