  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.
