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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.
MSC:
11B50 Sequences (mod \(m\))
11B83 Special sequences and polynomials
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References:
[1] D. Berend and Y. Bilu,Polynomials with roots modulo every integer, Proc. Amer. Math. Soc. 124 (1996), 1663-1671. · Zbl 1055.11523
[2] R. Graham, D. Knuth and O. Patashnik,Concrete Mathematics, 2nd ed., AddisonWesley, Reading, MA, 1994. · Zbl 0836.00001
[3] L. Hajdu, S. Laishram and S. Tengely,Power values of sums of products of consecutive integers, Acta Arith. 172 (2016), 333-349. · Zbl 1400.11087
[4] F. Luca,Prime divisors of binary holonomic sequences, Adv. Appl. Math. 40 (2008), 168-179. · Zbl 1165.11004
[5] T. Mansour,Combinatorics of Set Partitions, CRC Press, Boca Raton, FL, 2013. · Zbl 1261.05002
[6] P. Miska,Arithmetic properties of the sequence of derangements, J. Number Theory 163 (2016), 114-145. · Zbl 1405.11018
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