I am trying to understand how to go from a Butcher table to something useful of the form $y_{n+1} = g(y_n, y_{n+1})$.
Let us first try the simplest case, which is backwards Euler.
Using the coefficients from the table we would get:
$$k_1 = f(t +h, y_n + hk_1)$$ $$y_{n+1} = y_n + hk_1$$
In this setting we can do
$$y_{n+1} - y_n = hk_1 \iff k_1 = \frac{y_{n+1} - y_n}{h}$$
Replacing into the first equation yould yield:
$$k_1 = f(t+h, y_n + (y_{n+1} - y_n)) = f(t+h, y_{n+1})$$
And now into the second we get:
$$y_{n+1} = y_n + hf(t+h, y_{n+1})$$
How about in the more general setting? How do you replace each $k_i$ by some expersion of $y_{n+1}$? I am trying to come up with a good systematic way but the math is getting convoluted. Say, for example, we wanted to do third order RADUA IIA which has this table:
It makes no difference if the implicit equation is in $k_1$ or in $y_{n+1}$. And your observation is correct, with more than one stage you can not express the $k_1,k_2,...,k_s$ in $y_{n+1}$, but you can define the intermediate points $$y_{n,i}=y_n+h\sum_ja_{ij}k_j,~~k_i=f(t_n+c_ih,y_{n,i}),$$ and eliminate the slopes $k_i$ from the stages. I'm not aware of any numerical advantages of this substitution.