OrePolynomial[{{-t^3 + 4*t^5 - 6*t^9 + t^11 + 2*t^13, {4}}, 
  {4*(-2*t^6 - 13*t^8 + 8*t^10 + 7*t^12), {3}}, 
  {3*(t - 4*t^3 + 20*t^5 - 46*t^7 + 59*t^9 + 34*t^11), {2}}, 
  {3*(-1 + 4*t^2 + 28*t^4 - 34*t^6 + 69*t^8 + 30*t^10), {1}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
