Sums of products of binomial coefficients mod 2 and 2-regular sequences

TL;DR

This paper demonstrates that the run length transforms of certain linear recurrence sequences are 2-regular, enabling simplified proofs of sums of binomial coefficient products modulo 2 using computational tools.

Contribution

It establishes that run length transforms of these sequences are 2-regular, providing a new, computer-assisted approach to proving related combinatorial identities.

Findings

Run length transforms of these sequences are 2-regular.

Wu's results are recovered using Walnut software.

New identities are derived without lengthy proofs.

Abstract

Wu showed that certain sums of products of binomial coefficients modulo 2 are given by the run length transforms of several famous linear recurrence sequences, such as the positive integers, the Fibonacci numbers, the extended Lucas numbers, and Narayana's cows sequence. In this paper we show that the run length transform of such sequences are 2-regular sequences. This allows us to obtain Wu's results and some new ones using the computer program Walnut, eliminating the need for long technical proofs.