  We present novel arithmetic terms for univariate multinomial coefficients and
their partial sums. Notably, we introduce what appear to be the first closed
form expressions for partial sums of binomial coefficients. We extend our
findings to discover new arithmetic terms for multisections of binomial
coefficient sums. Applying our techniques to central trinomial coefficients, we
successfully resolve an open problem posed by Graham et al., by demonstrating
the existence of a "simple closed form" for these coefficients.
