OrePolynomial[{{-t^3 + 2*t^5 - 2*t^9 + t^11, {3}}, 
  {2*(-t^2 + 2*t^4 + 4*t^6 - 10*t^8 + 5*t^10), {2}}, 
  {5*t - 6*t^3 + 24*t^5 - 42*t^7 + 19*t^9, {1}}, 
  {3 - 18*t^2 + 56*t^4 - 30*t^6 + 5*t^8, {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
