OrePolynomial[{{216513*t^2 - 12275712*t^3 + 82354176*t^4 - 207618048*t^5 + 
    218103808*t^6, {4}}, {36*(24057*t - 1122768*t^2 + 10169344*t^3 - 
     39780352*t^4 + 54525952*t^5), {3}}, 
  {60*(8019 - 181116*t + 6043392*t^2 - 37896192*t^3 + 68157440*t^4), {2}}, 
  {120*(8019 + 321408*t - 6377472*t^2 + 13631488*t^3), {1}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
