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On a generalization of the recurrence defining the number of derangements. (English) Zbl 07262528
Summary: The sequence of derangements is given by the formula \(D_0=1,D_n=nD_{n-1}+(-1)^n,n > 0\). It is a classical object in combinatorics and number theory. We extend results on \((D_n)_{n\in\mathbb{N}}\) to a more general class of sequences given by the recurrence \(a_0=h_1(0),a_n=f(n)a_{n-1}+h_1(n)h_2(n)^n, n > 0\), where \(f,h_1,h_2\in\mathbb{Z}[X]\). We study arithmetic properties of these sequences, such as periodicity modulo \(d\in \mathbb{N}_+, p\)-adic valuations, rate of growth, periodicity, recurrence relations and prime divisors.
11B50 Sequences (mod \(m\))
11B83 Special sequences and polynomials
Full Text: DOI
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