منابع مشابه
The 2-adic Valuation of Stirling Numbers
We analyze properties of the 2-adic valuations of S(n, k), the Stirling numbers of the second kind. A conjecture that describes patterns of these valuations for fixed k and n modulo powers of 2 is presented. The conjecture is established for k = 5.
متن کاملThe 2-adic Valuation of Stirling Numbers of the Second Kind
In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v2(S(2n + 1, k + 1)) = s2(n) − 1 for any positive integer n, where s2(n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i 6≡ 7 (mod 32). This proves another conjecture of ...
متن کاملAlternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind
An interesting 2-adic property of the Stirling numbers of the second kind S(n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: ν2(S(2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n. It was later generalized to ν2(S(c2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n, c ≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on...
متن کاملOn the 2-adic order of Stirling numbers of the second kind and their differences
Let n and k be positive integers, d(k) and ν2(k) denote the number of ones in the binary representation of k and the highest power of two dividing k, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that ν2(S(2, k)) = d(k)−1, 1 ≤ k ≤ 2. Here we prove that ν2(S(c2, k)) = d(k)−1, 1 ≤ k ≤ 2, for any positive integer c. We improve and extend this statement in...
متن کاملOn 2-adic Orders of Stirling Numbers of the Second Kind
We prove that for any k = 1, . . . , 2 the 2-adic order of the Stirling number S(2, k) of the second kind is exactly d(k) − 1, where d(k) denotes the number of 1’s among the binary digits of k. This confirms a conjecture of Lengyel.
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ژورنال
عنوان ژورنال: Experimental Mathematics
سال: 2008
ISSN: 1058-6458,1944-950X
DOI: 10.1080/10586458.2008.10129026