I would look into script links. There’s some intro posts around on this forum (if you search) about getting started with them.
Its somewhat difficult to discuss this without going into painstaking detail, but basically, you just need to associate some piece of data (like your coefficients) with the frame number. Another option is to precompute it all, store it as a .obj file, and then use a script to import those .obj files on every frame (check out a script by the user Atom, called MeshFoot).
I would do some research into script links first off, and if it doesn’t help, we can try to give you some more direction!