There are well-known formulas for the coefficients resulting from multiplying two formal polynomials or generating functions (we define formal polynomial to be a generating function such that all coefficients of sufficiently high index are zero). Adding is even easier, as it is component-wise.
Question: What is the neatest formula for the coefficients of the composition $f\circ g$ where $f,g$ are formal polynomials or generating functions?
I tried approaching it by writing $(f\circ g)(x) = f(g(x))$ and substituting the expression for $g$ into the expression for $f$. Even with summation notation, the expansion just devolved into an mess involving the multinomial theorem that seems difficult to simplify. I am hopeful that combinatorial identities or other techniques such as exchanging the order of the summations will yield a somewhat satisfactory formula. I'll settle for pretty much any non-trivial simplification.
for OGF, let $[x^i] f(x) = f_i$ and $[x^j] g(x) = g_j$. Then you can rewrite the composition as $$ [x^k]f(g(x)) = \sum\limits_i f_i [x^k] g^i(x) = \boxed{\sum\limits_i f_i \sum\limits_{j_1 + \dots + j_i = k} g_{j_1} \dots g_{j_i}} $$
For EGF, let $\left[\frac{x^i}{i!}\right] f(x) = f_i$ and $\left[\frac{x^j}{j!}\right] g(x) = g_j$, then $$ \left[\frac{x^k}{k!}\right] f(g(x)) = \sum\limits_i \frac{f_i}{i!} \left[\frac{x^k}{k!}\right] g^i(x) = \boxed{\sum\limits_i \frac{f_i}{i!} \sum\limits_{j_1+\dots+j_i=k} \binom{k}{j_1,\dots,j_i} g_{j_1} \dots g_{j_i}} $$ I think it doesn't really get prettier than that, and it's indeed better to interpret combinatorially. Informally, if $f(x)$ is the genfunc of species $A$, and $g(x)$ is the genfunc of species $B$, then $f(g(x))$ is the genfunc of species obtained by taking instances of species $A$ and replacing their "atoms" by instances of species $B$.
For example, if $A$ is species of "sets" and $B$ is species of "cycles", then $f(g(x))$ is the genfunc for the species of "sets of cycles", aka "permutations". The meaning of the composition formula for EGFs is then interpreted as "take an instance of $A$ on $i$ atoms, and replace its atoms with instances of $B$ with total of $k$ atoms across them". Then the multinomial coefficient accounts for the number of ways to distribute $k$ new atoms between $i$ instances of species $B$.