  Despite the relevance of the binomial distribution for probability theory and
applied statistical inference, its higher-order moments are poorly understood.
The existing formulas are either not general enough, or not structured and
simplified enough for intended applications.
  This paper introduces novel formulas for binomial moments, in form of
\emph{polynomials in the variance} rather than in the success probability. The
obtained formulas are arguably better structured, simpler and superior in their
numerical properties compared to prior works. In addition, the paper presents
algorithms to derive these formulas along with working implementation in the
Python symbolic algebra package.
  The novel approach is a combinatorial argument coupled with clever algebraic
simplifications which rely on symmetrization theory. As an interesting
byproduct we establish \emph{asymptotically sharp estimates for central
binomial moments}, improving upon partial results from prior works.
