# Use of difference tables at studying of properties of binomial coefficients

## TL;DR

This paper investigates the properties of diagonal binomial coefficients, especially their units digits, using difference tables to predict digit appearances and explores their applications in modeling biological population dynamics.

## Contribution

It introduces a novel approach using difference tables to analyze units digit patterns in binomial coefficients and links these properties to biological modeling.

## Key findings

- Frequency of units digits shows gaps in common systems.
- Difference tables can predict digit occurrence in binomial coefficients.
- Binomial coefficient properties relate to cell population models.

## Abstract

Some properties of diagonal binomial coefficients were studied in respect to frequency of their units digits. An approach was formulated that led to use of difference tables to predict if certain units digits can appear in the values of binomial coefficients at quadratic terms of the binomial theorem. Frequency distributions of units digits of binomial coefficients contain gaps (zero frequency) under most common numbering systems with supposed exclusion to systems with 2^n bases. Binomial coefficient arithmetics may be used to model cell population dynamics in a multicellular organism, which implies that the dynamics obeys power function laws.