Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic Analysis of Central Binomiacci Numbers

Hebert Pérez-Rosés

TL;DR

The paper analyzes central Binomiacci numbers defined via a Fibonacci-boundary Pascal-like table. It derives univariate and bivariate generating functions, then applies Stanley's diagonal method to extract the diagonal generating function $C(s)$ and performs algebraic singularity analysis to obtain the asymptotic growth. The main result is $\mathcal{B}_{n,n} = \frac{3\cdot 4^n}{\sqrt{\pi n}} + o\left(\frac{4^n}{\sqrt{n}}\right)$, with the dominant singularity at $s=\tfrac{1}{4}$ and a removable nearby singularity at $s=\sqrt{5}-2$ that does not affect growth. Together, these findings connect Fibonacci-boundary combinatorics to diagonal extraction techniques for multivariate generating functions, providing explicit growth rates for the central terms of the sequence.

Abstract

We make an asymptotic analysis via singularity analysis of generating functions of a number sequence that involves the Fibonacci numbers and generalizes the binomial coefficients.

Asymptotic Analysis of Central Binomiacci Numbers

TL;DR

The paper analyzes central Binomiacci numbers defined via a Fibonacci-boundary Pascal-like table. It derives univariate and bivariate generating functions, then applies Stanley's diagonal method to extract the diagonal generating function and performs algebraic singularity analysis to obtain the asymptotic growth. The main result is , with the dominant singularity at and a removable nearby singularity at that does not affect growth. Together, these findings connect Fibonacci-boundary combinatorics to diagonal extraction techniques for multivariate generating functions, providing explicit growth rates for the central terms of the sequence.

Abstract

We make an asymptotic analysis via singularity analysis of generating functions of a number sequence that involves the Fibonacci numbers and generalizes the binomial coefficients.

Paper Structure

This paper contains 3 sections, 4 theorems, 22 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Generating functions
  3. Asymptotic Analysis

Key Result

lemma 1

Let $A_k(z) = \sum_{n \geq 0} \mathcal{B}_{k,n} z^n$ be the univariate ordinary generating function associated with the sequence $\langle \mathcal{B}_{k,n} \rangle$, where $z \in \mathbb{C}$, and let $G(z,w) = \sum_{k \geq 0} \sum_{n \geq 0} \mathcal{B}_{k,n} z^n w^k$ be the corresponding bivariate and

Theorems & Definitions (6)

  • lemma 1
  • proof
  • theorem 1
  • proof
  • theorem 2
  • corollary 1