On sums related to central binomial and trinomial coefficients

TL;DR

This paper explores congruences and series involving central binomial and trinomial coefficients, proposing new conjectures and series for 1/π based on prime representations and quadratic forms.

Contribution

It introduces numerous conjectures on prime-related congruences and presents 62 new series for 1/π inspired by these congruences and dualities.

Findings

Numerous conjectures on prime representations by quadratic forms

62 new series for 1/π proposed

Insights into congruences related to binomial and trinomial coefficients

Abstract

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic forms, and 62 proposed new series for $1/\pi$ motivated by congruences and related dualities.