(This question comes from the last input of Converting Equations to Sparse Arrays in Wolfram Documentation.)

I want to get this (by extensible code):

s0 + s1.{x,y} + s2.{x,y}.{x,y} + s3.{x,y}.{x,y}.{x,y} 

But this is my failed trial:

In:

Symbol["s" <> ToString[#]] & /@ Range[0, 3]  (*Focus below:*)  NestList[HoldForm@Dot[#, {x, y}] &, {x, y}, 2]~Prepend~1 Inner[Times, %%, %, Plus]~Collect~{x, y} 

Out:

{s0, s1, s2, s3} {1,{x,y},{x,y}.{x,y},({x,y}.{x,y}).{x,y}} {s0+s1 x+s3 ({x,y}.{x,y}).{x,y}+s2 {x,y}.{x,y},s0+s1 y+s3 ({x,y}.{x,y}).{x,y}+s2 {x,y}.{x,y}} 

Problems

1. Parentheses are redundant.

2. I don’t know how to apply MatrixPower to achieve that, since the documentation says “MatrixPower works only on square matrices.”

Thanks for help!