We describe the combinatorics that arise in summing a double recursion formula for the enumeration of connected Feynman graphs in quantum field theory. In one index the problem is more tractable and yields concise formulas which are combinatorially interesting on their own. In the other index, one of these sums is Sloane's sequence A001865.
We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences.
The Kolakoski sequence $S$ is the unique element of $\left\lbrace 1,2 \right\rbrace^ω$ starting with 1 and coinciding with its own run length encoding. We use the parity of the lengths of particular subclasses of initial words of $S$ as a unifying tool to address the links between the main open questions - recurrence, mirror/reversal invariance and asymptotic density of digits. In particular we prove that recurrence implies reversal invariance, and give sufficient conditions which would imply that the density of 1s is $\frac{1}{2}$.
A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with positive coefficients and a particular set of initial conditions. A sequence of positive integers is \emph{complete} if every positive integer is a sum of distinct terms of the sequence. One consequence of Zeckendorf's theorem is that the sequence of Fibonacci numbers is complete. Previous work has established a generalized Zeckendorf's theorem for all PLRS's. We consider PLRS's and want to classify them as complete or not. We study how completeness is affected by modifying the r…
We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n-L+1}$ and a particular set of initial conditions. We seek to classify various PLRS's by completeness. With results on how completeness is affected by modifying the recurrence coefficients of a PLRS, we completely characterize completeness of several families of PLRS's as well as conjecturing criteria for more…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward descriptions in terms of how often each value in the sequence occurs. In this paper, we generalize the most classical examples to a larger family of sequences parametrized by linear recurrence relations. Each of our sequences can be constructed in three different ways: via a nested recurrence relation, from labeled infinite trees, or by using Zeckendorf-like strin…
A symbolic method for solving linear recurrences of combinatorial and statistical interest is introduced. This method essentially relies on a representation of polynomial sequences as moments of a symbol that looks as the framework of a random variable with no reference to any probability space. We give several examples of applications and state an explicit form for the class of linear recurrences involving Sheffer sequences satisfying a special initial condition. The results here presented can be easily implemented in a symbolic software.
We propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be deduced from the Skolem-Mahler-Lech theorem and the Subspace Theorem. We will give details of this deduction that is due to P. Corvaja in the full version of this extended abstract.
We introduce the "moment rank" and "unitary rank" of numerical sequences, close relatives of linear-recursive order. We show that both parameters can be characterized by a broad set of criteria involving moments of measures, types of recurrence relations, Hankel matrix factorizations, Waring rank, analytic properties of generating functions, and algebraic properties of polynomial ideals. In the process, we solve the "complex finite-atomic" and "integral finite-atomic" moment problems: which sequences arise as the moments of a finite-atomic complex-/integer-valued measures on $\mathbb{C}$?
Let $V(k)$ denote the waiting time, the number of trials needed to get a consecutive $k$ ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of $V(k)$ in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order $k$. Maximum likelihood estimation (MLE) methods for the probability distributions are presented in both cases with simulation examples.
