Define {f(n)}n=1, the floor sequence, by the linear recurrence f(n + 1) = n ∑ k=1 ⌊n k ⌋ f(k), f(1) = 1. Similarly, define {g(n)}n=1, the roof sequence, by the linear recurrence g(n + 1) = n ∑ k=1 ⌈n k ⌉ g(k), g(1) = 1. This paper studies various properties of these two sequences, including prime criteria, asymptotic approximations of { f(n+1) f(n) }∞ n=1 and { g(n+1) g(n) }∞ n=1 , and the iter...

We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials Pn(z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments Mq(Pn) of even order q for q 32. We were also able to check a conjecture on the asymptotic behavior of Mq(Pn), namely Mq(Pn) ∼ Cq2, where Cq = 2...

An ordered partition of [n] = {1, 2, . . . , n} is a partition whose blocks are endowed with a linear order. Let OPn,k be the set of ordered partitions of [n] with k blocks and OPn,k(σ) be the set of ordered partitions in OPn,k that avoid a pattern σ. For any permutation pattern σ of length three, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [n] wi...

Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, downloa...

Many algorithms for computing the reliability of linear or circular consecutive-k-out-of-n:F systems appeared in this Transactions. The best complexity estimate obtained for solving this problem is O(k log(n/k)) operations in the case of i.i.d. components. Using fast algorithms for computing a selected term of a linear recurrence with constant coefficients, we provide an algorithm having arithm...

We present an algorithm which decides the shift equivalence problem for Pfinite sequences. A sequence is called P-finite if it satisfies a homogeneous linear recurrence equation with polynomial coefficients. Two sequences are called shift equivalent if shifting one of the sequences s times makes it identical to the other, for some integer s. Our algorithm computes, for any two P-finite sequence...

We prove that for arbitrary partitions λ ⊆ κ, and integers 0 6 c < r 6 n, the sequence of Schur polynomials S(κ+k·1c)/(λ+k·1r)(x1, . . . , xn) for k sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom’s determin...

We show that multiple orthogonal polynomials for r measures (μ1, . . . , μr) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices ~n ± ~ej, where ~ej are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal ...

Convergence of a matrix dynamics for online LDA is analyzed. Especially, stable spurious solutions are pointed out and two schemes to prevent the spurious solutions are proposed. The performance of the algorithm is confirmed by simulations of face identification.