Explicit formulas using partitions of integers for numbers defined by recursion

TL;DR

This paper derives explicit formulas for sequences defined by recursion using integer partitions, providing a purely arithmetic proof and comparing with historical approaches, with applications to Bernoulli, Euler, and Fibonacci numbers.

Contribution

It introduces a new explicit formula for recursive sequences based on integer partitions, using only arithmetic methods, and connects with classical formulas from the 19th century.

Findings

Explicit formulas for recursive sequences using partitions

Comparison with historical formulas from the 19th century

Applications to Bernoulli, Euler, and Fibonacci numbers

Abstract

In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare our result with similar formulas obtained with different approaches already in the XIX century. Examples are given for Bernoulli, Euler and Fibonacci numbers.