Given $f(t)=a_0+a_1t+a_2t^2+...$, it is straightforward to express $a_0+a_1ut+a_2u^2t^2+...$ through $f$ - it is just $f(ut)$ of course.
What I need is to express through $f$ the series$$a_0+a_1t+a_2t^2+a_3u^3t^3+a_4u^3t^3+a_5u^3t^3+a_6u^6t^6+a_7u^6t^6+...$$
What I managed is$$\frac{1-u^3}{3u^2}\left(\frac1{1-u}f(ut)+\frac\omega{1-\omega u}f(\omega ut)+\frac{\omega^2}{1-\omega^2u}f(\omega^2ut)\right)$$where $\omega$ is the third degree root of unity.
Can one do better? That is, simpler?