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We study two families of sequences, listed in the On-Line Encyclopedia ofInteger Sequences (OEIS), which are associated to invariant theory of Liealgebras. For the first family, we prove combinatorially that the sequencesA059710 and A108307 are related by a binomial transform. Based on this, wepresent two independent proofs of a recurrence equation for A059710, which wasconjectured by Mihailovs. Besides, we also give a direct proof of Mihailovs'conjecture by the method of algebraic residues. As a consequence, closedformulae for the generating function of sequence A059710 are obtained in termsof classical Gaussian hypergeometric functions. Moreover, we show thatsequences in the second family are also related by binomial transforms.

On sequences associated to the invariant theory of rank two simple Lie algebras