On the lcm-analog of binomial coefficient. (English) Zbl 1384.11036
Summary: Let \(n\geq1\) and \(k\geq0\) be integers with \(n\geq k\). Let \(C(n,k)\) denote the binomial coefficient indexed by \(n\) and \(k\), i.e. \(C(n,k)=\frac{n\times(n-1)\times\dots\times(n-k+1)} {1\times2\times\dots\times k}\). Define the lcm-analog of binomial coefficient by \(L(n,k)\frac{\mathrm {lcm}(n,n-1,\dots,n-k+1)} {\mathrm {lcm}(1,2,\dots,k)}\). In this paper, we prove that \(L(n,n-k)\) divides \(L(n,k)\) if \(k\leq\frac n2\), and give some couples \((n,k)\) satisfying that \(L(n,k)=L(n,n-k)\). Subsequently, we give a sufficient condition such that \(L(n,k)\) divides \(L(n+1,k)\). Finally, we provide a new proof for the known fact that \(L(n,k)\) divides \(C(n,k)\), and present many couples \((n,k)\) such that \(L(n,k)=C(n,k)\).
MSC:
11B65 | Binomial coefficients; factorials; \(q\)-identities |
11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |
Online Encyclopedia of Integer Sequences:
Triangle read by rows: T(n,k) = lcm(n, n-1, ..., n-k+2, n-k+1)/lcm(1, 2, ..., k) (1 <= k <= n).References:
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