Sums of products of binomial coefficients mod 2 and run length transforms of sequences

TL;DR

This paper investigates properties of binomial coefficients mod 2, deriving recurrence relations that connect sums of their products to run length transforms of well-known sequences like Fibonacci and Lucas numbers.

Contribution

It introduces new recurrence relations for sums of products of binomial coefficients mod 2 and links these to run length transforms of classical sequences.

Findings

Derived recurrence relations for sums of products of binomial coefficients mod 2

Connected these sums to run length transforms of key sequences

Provided explicit formulas for transforms of Fibonacci, Lucas, and Narayana sequences

Abstract

We study properties of functions of binomial coefficients mod 2 and derive a set of recurrence relations for sums of products of binomial coefficients mod 2. We show that they result in sequences that are the run length transforms of well known basic sequences. In particular, we obtain formulas for the run length transform of the positive integers, Fibonacci numbers, extended Lucas numbers and Narayana's cows sequence.