OrePolynomial[{{t^3 - 2*t^7 + t^11, {3}}, 
  {2*(-t^2 - 3*t^4 - 3*t^6 + 3*t^8 + 4*t^10), {2}}, 
  {3*t + 6*t^3 - 8*t^5 + 18*t^7 + 13*t^9, {1}}, 
  {3*(-1 - 2*t^2 + 2*t^6 + t^8), {0}}}, OreAlgebraObject[{Der[t]}, Together, 
  Together[#1 + #2] & , Together[#1*#2] & , None], DegreeLexicographic]
