On sums related to central binomial and trinomial coefficients
Zhi-Wei Sun

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 is the coefficient of in the expansion of with . 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 motivated by congruences and related dualities.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · History and Theory of Mathematics
