The pdf below presents relations among compositional and multiplicative inversions, Laurent series, and the formal free cumulants and moments of free probability. One example is the q-expansion of Klein’s j-invariant found in monster moonshine. Another is based on Motzkin numbers.
More Related Stuff:
“Mastering the Master Field” by Gopakumar and Gross, pp. 17-19.
“Large N Gauge Theory – Expansions and Transitions” by Douglas, discussions of free probability.
Shadows of Simplicity 
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