A Method for Obtaining Generating Function for Central Coefficients of Triangles

TL;DR

This paper introduces techniques to derive generating functions for the central coefficients of combinatorial triangles, providing solutions for direct and inverse problems related to these generating functions.

Contribution

It presents new methods for obtaining generating functions for central coefficients of triangles and proves theorems for solving related direct and inverse problems.

Findings

Derived explicit generating functions for central coefficients

Established theorems for solving direct problems

Developed methods for inverse problem solutions

Abstract

We present techniques for obtaining a generating function for the central coefficients of a triangle $T(n,k)$, which is given by the expression $[xH(x)]^k=\sum_{n\geqslant k} T(n,k)x^n$, $H(0)\neq 0$. We also prove certain theorems for solving direct and inverse problems.