Let C_n,k:=fracn-2knn choose k=n-1 choose k-n-1 choose k-1, with the Catalan triangle relation B_n,k=C_2n,n-k. Let n and a be positive integers with n>a, and let r be a non-negative integer. Miana–Ohtsuka–Romero's conjecture. The sum of odd powers satisfies ∑_k=0^aC_n,k^2r+1equiv 0 pmodn-1 choose a. This is the second conjecture announced in the paper, whose abstract says that it is proved by establishing a q-analogue; the source develops the proof through the stated congruence and related polynomial results.

Miana–Ohtsuka–Romero's Catalan triangle odd-power congruence

About 8 years old · traced to 2018

Let $C_{n,k}:=\frac{n-2k}{n}{n\choose k}={n-1\choose k}-{n-1\choose k-1}$ , with the Catalan triangle relation $B_{n,k}=C_{2n,n-k}$ . Let $n$ and $a$ be positive integers with $n>a$ , and let $r$ be a non-negative integer. Miana–Ohtsuka–Romero's conjecture. The sum of odd powers satisfies

\sum_{k=0}^{a}C_{n,k}^{2r+1}\equiv 0\pmod{{n-1\choose a}}.

This is the second conjecture announced in the paper, whose abstract says that it is proved by establishing a $q$ -analogue; the source develops the proof through the stated congruence and related polynomial results.

References

Primary source

Victor J. W. Guo and Xiuguo Lian, “Proofs of two conjectures on Catalan triangle numbers”, arXiv:1806.02685 (2018).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2018 paper claims to prove the conjecture using a stronger polynomial version, but the proof has not been independently verified here.

Miana, Ohtsuka, and Romero conjectured that the sum of the odd powers of the first $a+1$ Catalan-triangle entries is divisible by the relevant binomial coefficient for all allowed positive integers and non-negative powers.

Known results

Earlier work by Guo and Zeng, and by Guo and Wang, established only special cases, according to the later paper.

2018 claimed proof

The paper states that it confirms the conjecture by proving $q$ -analogues for the associated Catalan-triangle sums. Specializing $q=1$ yields the stated congruence and equivalent congruences for $B_{n,k}$ and $A_{n,k}$ . No retrieved source reports a counterexample, correction, withdrawal, or dispute.

Current status (as of September 2026): The conjecture is claimed proved by the 2018 $q$ -analogue paper, but that resolution remains unverified in this record.

Sources

No solutions have been posted yet.