OrePolynomial[{{-7*t^3 + 8*t^5 + 20*t^7 - 18*t^9 - 17*t^11 + 10*t^13 + 
    4*t^15, {4}}, {4*(10*t^4 + 6*t^6 - 47*t^8 - 24*t^10 + 43*t^12 + 12*t^14), 
   {3}}, {3*(7*t + 24*t^3 + 72*t^5 - 170*t^7 + 21*t^9 + 194*t^11 + 44*t^13), 
   {2}}, {3*(-7 - 24*t^2 - 24*t^4 - 166*t^6 + 11*t^8 + 94*t^10 + 20*t^12), 
   {1}}}, OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
