The sum $\sum_{k=0}^{q-1}\binom{2k}{k}$ for q a power of 3

TL;DR

This paper proves a conjecture relating a sum of central binomial coefficients to powers of 3, revealing a modular invariance property and providing an efficient computation method for the sum's normalized value.

Contribution

It establishes a precise modular congruence for the sum of binomial coefficients when q is a power of 3, confirming a conjecture and extending understanding of these sums.

Findings

Sum $inom{2k}{k}$ for k=0 to q-1 is congruent to q^2 modulo 3q^2.

Normalized sum (1/q^2) times the sum is independent of q for large q.

Provides an efficient method to compute the normalized sum modulo powers of 3.

Abstract

We prove that $\sum_{k=0}^{q-1}\binom{2k}{k}\equiv q^2\pmod{3q^2}$ if q>1 is a power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our more precise result actually implies that the value of $(1/q^2)\sum_{k=0}^{q-1}\binom{2k}{k}$ modulo a fixed arbitrary power of 3 is independent of q, for q a power of 3 large enough, and shows how such value can be efficiently computed.