Yes, look for "Deligne's exceptional series". There are no theorems, but several beautiful conjectures.
SeeThe basic idea is that there should be a symmetric pivotal category generated by a trivalent vertex, forwith just a startfew local relations, Le série exceptionnelle de groupes dedepending on a parameter. At special values of the parameter, the category becomes degenerate, and the quotient by the negligible ideal recovers the representation category of one of the exceptional Lie algebras. - Deligne P(More or less; in some cases you get an equivariantization or subcategory.)
Working over rational functions in the parameter instead, it is expected that the category is semisimple, and its moments should agree with the sequence you describe. The exceptional algebra - C$F_4$ is the `least degenerate' point, in that its moments fall short the least. R
Here are some pointers to the literature. Acad
Pierre Deligne, La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. Sci4, 321--326. Paris Sér
Pierre Deligne and Ronald de Man, La série exceptionnelle de groupes de Lie. II, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. I Math6, 577--582. 322
Arjeh M. Cohen and Ronald de Man, Computational evidence for Deligne’s conjecture regarding exceptional Lie groups, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 45, 321–326427--432.
Pierre Deligne and Benedict H. Gross, On the exceptional series, and its descendants, C. R. Math. Acad. Sci. Paris 335 (2002), no. 11, 877- MR1378507-881.
J. M. Landsberg and L. Manivel, and follow citations forwards and backwards!Series of Lie groups, Michigan Math. J. 52 (2004), no. 2, 453--479.
J. M. Landsberg and L. Manivel, Triality, exceptional Lie algebras and Deligne dimension formulas, Adv. Math. 171 (2002), no. 1, 59--85.