I have encountered an equality $$ \sum_{n=0}^\infty p_{3n}(t)\left(\frac t{(1-t)(1-2t)}\right)^n=\frac{1-2t}{1-4t}. $$ I know that $p_0(t)=1$, $p_3(t)=2(1-t)^2$, while for $n>1$, $p_{3n}(t)$ is a polynomial in $t$ of degree $3n$ with integer coefficients, with leading term $-4^{n-1}t^{3n}$ and divisible by $2(1-t)^2$.
How much can be said about these polynomials given the equality? Specifically, I need to determine $p_{3n}(0)$. Most likely, the above equality does not suffice to do this, what kind of additional information could you think of which would help to find the constant terms?
I've never seen problems like this before, I don't even know what are appropriate tags.