Interpolation Attack
As in the first equation one has $(x+k_1)^3 = (x^2+k_1^2)(x+k_1)$, i.e., $(x+k_1)^2 = x^2+k_1^2$ I'd conclude that the equation is over characteristic $2$.

This explains why all the terms with even coefficients are missing and why all odd coefficients are replaced by $1$.

The only remaining mystery is what happened to $3k_1^6k_2$ rsp. $k_1^6k_2$. Typo?