Binomial self-inverse sequences and tangent coefficients

AbstractThis paper treats the class of sequences {an} that satisfy the recurrence relation a2n+1=∑k=0n(−1)k(nkakdn−k between the odd and even terms of {an} that involves the coefficients of tan(t), namely a2n+1=∑k=0n(−1)k(2n+12k+1)Tk(d/2)2k+1a2n−2kA combinatorial setting is then provided to elucidate the appearance of the tangent coefficients in this equation

On some properties of the number of permutations being products of pairwise disjoint d-cycles

Let d≥2 be an integer. In this paper we study arithmetic properties of the sequence (Hd(n))n∈N, where Hd(n) is the number of permutations in Sn being products of pairwise disjoint cycles of a fixed length d. In particular we deal with periodicity modulo a given positive integer, behaviour of the p-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study their properties. Among many results we obtain qualitative description of the p-adic valuation of the number Hd(n) extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida