Let $p$ be a prime and suppose that $g(t)$ is the generating function $g(t) = 1/p\,g(t)^p + t$ with low order terms $g(t) = t + O(t^p)$. An easy induction shows that the coefficient of $t^n$ in $g(t)$ is zero unless $n\equiv 1\pmod{p-1}$. I need to show that the $p$-adic valuation of $[t^n] g(t)$ is equal to $-v(n!)$ when $n\equiv 1\pmod{p-1}$.

Expanded out, the recurrence says that

$\displaystyle f(n) = 1/p \sum_{a_1+\cdots + a_p = n} \prod_{i} f(i)$

When $p=3$, this sequence starts

$\displaystyle t+\frac{t^3}{3}+\frac{t^5}{3}+\frac{4 \, t^7}{9}+\frac{55 \,t^9}{81}+\frac{91 \,t^{11}}{81}+\frac{476 \,t^{13}}{243}+\frac{2584 \,t^{15}}{729}+\frac{4807 \,t^{17}}{729}+O(t^{19})$.

Does anyone have any suggestions for how to check this?