# An asymptotic Formula for the iterated exponential Bell Numbers

## TL;DR

This paper derives an asymptotic formula for the leading coefficient of the iterated exponential Bell numbers, advancing understanding of their growth and structure.

## Contribution

It provides the first explicit asymptotic formula for the leading coefficients of Bell's iterated exponential numbers.

## Key findings

- Derived an asymptotic expression for the leading coefficient
- Enhanced understanding of the growth rate of iterated exponential Bell numbers
- Established a foundation for further coefficient analysis

## Abstract

In 1938 E. T. Bell introduced "The Iterated Exponential Integers". He proved that these numbers may be expressed by polynomials with rational coefficients. However, Bell gave no formulas for any of the coefficients except the trivial one, which is always 1. Our task has been to find the coefficient of the leading term, giving asymptotic information about these numbers.