  We recall some basic properties of the Beta distribution and some of its
modifications. We identified around $20$ of the moment sequences of Beta
distributions as important integer sequences in the OEIS base of integer
sequences. Among those identified are Catalan, Riordan, Motzkin, or 'super
ballot numbers'. By applying a method of expansion of the ratio of densities of
involved distributions we are able to obtain some known and many unknown
relationships between e.g. Catalan numbers and other moment sequences of the
Beta distributions.
