Related papers: Woon's tree and sums over compositions

We apply a tree-based methodology to solve new, very broadly defined families of nested recursions of the general form R(n)=sum_{i=1}^k R(n-a_i-sum_{j=1}^p R(n-b_{ij})), where a_i are integers, b_{ij} are natural numbers, and k,p are…

We discuss algebraic universality in the sense of P. Vogel for the simplest refined quantity, the Macdonald dimensions. The main known source of universal quantities is given by Chern-Simons theory. Refinement of Chern-Simons theory means…

The perturbation expansion of the solution of a fixed point equation or of an ordinary differential equation may be expressed as a power series in the perturbation parameter. The terms in this series are indexed by rooted trees and depend…

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which…

We consider weighted generating functions of trees where the weights are products of functions of the sizes of the subtrees. This work begins with the observation that three different communities, largely independently, found substantially…

Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of…

We study the properties of the third order sequence $(w_n)=\left(w_n(a,b,c; r, s,t)\right)$ defined by the recurrence relation $w_n = rw_{n - 1} + sw_{n - 2} + tw_{n - 3}\, (n \ge 3)$ with $w_0 = a,\,w_1 = b,\,w_2=c$, where $a$, $b$, $c$,…

We study Birkhoff sums over rotations (series of the form $\sum_{r=1}^{N}\phi(r\alpha)$), in which the summed function $\phi$ may be unbounded at the origin. Estimates of these sums have been of significant interest and application in pure…

We provide a new derivation of the well-known generating function counting the number of walks on a regular tree that start and end at the same vertex, and more generally, a generating function for the number of walks that end at a vertex a…

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…

In this paper, we consider infinite sums derived from the reciprocals of the generalized Fibonacci numbers. We obtain some new and interesting identities for the generalized Fibonacci numbers.

Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…

Recursive Neural Network (RecNN), a type of models which compose words or phrases recursively over syntactic tree structures, has been proven to have superior ability to obtain sentence representation for a variety of NLP tasks. However,…

The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due…

We define certain natural finite sums of $n$'th roots of unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n…

We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…

Every binary De~Bruijn sequence of order n satisfies a recursion 0=x_n+x_0+g(x_{n-1}, ..., x_1). Given a function f on (n-1) bits, let N(f; r) be the number of functions generating a De Bruijn sequence of order n which are obtained by…

We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly…

Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…