Finding the pattern of Lagrange Inversion Formula for $x+\sin x$

Question

Today I read about the Lagrange Inversion Theorem, which to restate: $$f\text{ is analytic at point }a, f'(a) \neq 0 \\ \implies f^{-1}(x)=a+\sum_{n=1}^\infty g_n\frac{(x-f(a))^n}{n!}, \\ g_n=\lim_{t\rightarrow a}\bigg(\frac{d^{n-1}}{dt^{n-1}}\bigg[\Big(\frac{t-a}{f(t)-f(a)}\Big)^n\bigg]\bigg)$$ To test it, I chose $f(x)=x+\sin x$ and $a=0$. The formula then became: $$f^{-1}(x)=\sum_{n=1}^\infty g_n\frac{x^n}{n!},g_n=\lim_{t\rightarrow 0}\bigg(\frac{d^{n-1}}{dt^{n-1}}\bigg[\Big(\frac{t}{t+\sin t}\Big)^n\bigg]\bigg)$$ I computed $g_n$ for small $n$ first. The first odd ones include $$g_1=\frac{1}{2},g_3=\frac{1}{16},g_5=\frac{1}{16},g_7=\frac{43}{256},g_9=\frac{223}{256},g_{11}=\frac{60623}{8192},g_{13}=\frac{764783}{8192};$$ {$g_2,g_4,\dots,g_{14}$} have all shown to be zero. The only noticeable pattern I can see in $g_n$ for odd n is that for the denominators, which I'll call $d_n$, is that $d_{n+4}=2^{(n+11)/4}d_n$; I cannot see a pattern for the numerators. How do I find the formula for the coefficients? Or does no closed formula (other than the limit above) exist?