Yes, look for "Deligne's exceptional series". There are no theorems, but several beautiful conjectures.

The basic idea is that there should be a symmetric pivotal category generated by a trivalent vertex, with just a few local relations, depending 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. (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 $F_4$ is the `least degenerate' point, in that its moments fall short the least.

Here are some pointers to the literature.

<cite authors="Pierre Deligne" mrnumber="1378507" cite="_C. R. Acad. Sci. Paris Sér. I Math._ **322** (1996), no. 4, 321--326">_Pierre Deligne_, [**La série exceptionnelle de groupes de Lie**](http://www.ams.org/mathscinet-getitem?mr=1378507), _C. R. Acad. Sci. Paris Sér. I Math._ **322** (1996), no. 4, 321--326.</cite>

<cite authors="Pierre Deligne and Ronald de Man" mrnumber="1411045" cite="_C. R. Acad. Sci. Paris Sér. I Math._ **323** (1996), no. 6, 577--582">_Pierre Deligne and Ronald de Man_, [**La série exceptionnelle de groupes de Lie. II**](http://www.ams.org/mathscinet-getitem?mr=1411045), _C. R. Acad. Sci. Paris Sér. I Math._ **323** (1996), no. 6, 577--582.</cite>

<cite authors="Arjeh M. Cohen and Ronald de Man" mrnumber="1381778" cite="_C. R. Acad. Sci. Paris Sér. I Math._ **322** (1996), no. 5, 427--432">_Arjeh M. Cohen and Ronald de Man_, [**Computational evidence for Deligne’s conjecture regarding exceptional Lie groups**](http://www.ams.org/mathscinet-getitem?mr=1381778), _C. R. Acad. Sci. Paris Sér. I Math._ **322** (1996), no. 5, 427--432.</cite>

<cite authors="Pierre Deligne and Benedict H. Gross" mrnumber="1952563" cite="_C. R. Math. Acad. Sci. Paris_ **335** (2002), no. 11, 877--881">_Pierre Deligne and Benedict H. Gross_, [**On the exceptional series, and its descendants**](http://dx.doi.org/10.1016/S1631-073X(02)02590-6), _C. R. Math. Acad. Sci. Paris_ **335** (2002), no. 11, 877--881.</cite>

<cite authors="J. M. Landsberg and L. Manivel" mrnumber="2069810" cite="_Michigan Math. J._ **52** (2004), no. 2, 453--479">_J. M. Landsberg and L. Manivel_, [**Series of Lie groups**](http://dx.doi.org/10.1307/mmj/1091112085), _Michigan Math. J._ **52** (2004), no. 2, 453--479.</cite>

<cite authors="J. M. Landsberg and L. Manivel" mrnumber="1933384" cite="_Adv. Math._ **171** (2002), no. 1, 59--85">_J. M. Landsberg and L. Manivel_, [**Triality, exceptional Lie algebras and Deligne dimension formulas**](http://dx.doi.org/10.1006/aima.2002.2071), _Adv. Math._ **171** (2002), no. 1, 59--85.</cite>