OrePolynomial[{{2*t^3 - 7*t^5 + 5*t^7 + 5*t^9 - 7*t^11 + 2*t^13, {4}}, 
  {4*(t^2 - 5*t^4 + 6*t^6 + 9*t^8 - 16*t^10 + 5*t^12), {3}}, 
  {3*(-6*t + 11*t^3 - t^5 + 11*t^7 - 33*t^9 + 10*t^11), {2}}, 
  {-3*(-6 + 11*t^2 + 63*t^4 - 53*t^6 - 17*t^8 + 10*t^10), {1}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
