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Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind
Published/Copyright:
September 9, 2010
Abstract
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(2n, k)) = d2(k) – 1, 1 ≤ k ≤ 2n. It was later generalized to ν2(S(c2n, 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 non-standard recurrence relations for S(n, k) in the second parameter and congruential identities.
Received: 2010-01-29
Accepted: 2010-05-04
Published Online: 2010-09-09
Published in Print: 2010-September
© de Gruyter 2010
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Articles in the same Issue
- Two New Van Der Waerden Numbers: w(2; 3, 17) and w(2; 3, 18)
- Distance Graphs from p-adic Norms
- A Note on Stirling Series
- On the Number of Certain Relatively Prime Subsets of {1, 2, . . . , n}
- Powers of Sierpiński Numbers Base b
- Subsets of ℤ with Simultaneous Orderings
- Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind
- On Relatively Prime Sets Counting Functions
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Keywords for this article
Stirling numbers of the second kind;
congruences and divisibility;
Bernoulli numbers
Articles in the same Issue
- Two New Van Der Waerden Numbers: w(2; 3, 17) and w(2; 3, 18)
- Distance Graphs from p-adic Norms
- A Note on Stirling Series
- On the Number of Certain Relatively Prime Subsets of {1, 2, . . . , n}
- Powers of Sierpiński Numbers Base b
- Subsets of ℤ with Simultaneous Orderings
- Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind
- On Relatively Prime Sets Counting Functions
- Congruences for Overpartition k-tuples
