  We study two families of sequences, listed in the On-Line Encyclopedia of
Integer Sequences (OEIS), which are associated to invariant theory of Lie
algebras. For the first family, we prove combinatorially that the sequences
A059710 and A108307 are related by a binomial transform. Based on this, we
present two independent proofs of a recurrence equation for A059710, which was
conjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs'
conjecture by the method of algebraic residues. As a consequence, closed
formulae for the generating function of sequence A059710 are obtained in terms
of classical Gaussian hypergeometric functions. Moreover, we show that
sequences in the second family are also related by binomial transforms.

Combinatorics;Symbolic Computation