Given the inverses of two Toeplitz matrices is there a way to get the first row of the inverse of their sum without doing another inverse

Question

I have a calculation that requires the inverse of the weighted sum of two Toeplitz matrices aT1 + bT2 for many different weights a,b. Inverting T1, T2 or aT1+bT2 is an O(n^2) operation using Levinson/Trench. I have to repeat that for each a,b and that is the dominant effort in my problem. So I would like to find a way of inverting T1 and T2 separately once and for all and then somehow with less than O(n^2) effort get the first row of the inverse of aT1 +bT2 for each a,b