Congruences for partial sums of the generating series for 3kk

We produce congruences modulo a prime [Formula: see text] for sums [Formula: see text] over ranges [Formula: see text] and [Formula: see text], where [Formula: see text] is a power of [Formula: see text]. Here [Formula: see text] equals either [Formula: see text], or [Formula: see text], where [Formula: see text] and [Formula: see text] are indeterminates. In the former case, we deal more generally with shifted binomial coefficients [Formula: see text]. Our method derives such congruences directly from closed forms for the corresponding series.