On sums related to central binomial and trinomial coefficients

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 $1\mathrm{/}\pi$ motivated by congruences and related dualities.