OrePolynomial[{{-3*t^3 - 16*t^4 - 37*t^5 - 226*t^6 + 822*t^7 + 3608*t^8 + 
    2584*t^9 + 19072*t^10 - 38784*t^11 - 169984*t^12 - 13824*t^13 - 
    294912*t^14 - 98304*t^15 + 294912*t^16, {5}}, 
  {2*(-21*t^2 - 86*t^3 - 21*t^4 - 245*t^5 + 9932*t^6 + 34644*t^7 + 
     19304*t^8 + 100864*t^9 - 654144*t^10 - 2319872*t^11 - 697344*t^12 - 
     3059712*t^13 - 1671168*t^14 + 3686400*t^15), {4}}, 
  {4*(-39*t - 68*t^2 + 786*t^3 + 3856*t^4 + 29732*t^5 + 86040*t^6 - 
     28552*t^7 - 151520*t^8 - 3347904*t^9 - 9929216*t^10 - 4799616*t^11 - 
     9897984*t^12 - 8454144*t^13 + 14745600*t^14), {3}}, 
  {24*(-6 + 30*t + 473*t^2 + 2172*t^3 + 7144*t^4 + 12956*t^5 - 83716*t^6 - 
     353552*t^7 - 2030496*t^8 - 5235328*t^9 - 3342528*t^10 - 3861504*t^11 - 
     5111808*t^12 + 7372800*t^13), {2}}, 
  {24*(36 + 291*t + 810*t^2 - 1714*t^3 - 19344*t^4 - 159376*t^5 - 
     629536*t^6 - 2341824*t^7 - 5463040*t^8 - 4204416*t^9 - 2856960*t^10 - 
     5996544*t^11 + 7372800*t^12), {1}}, 
  {48*(-6 - 159*t - 1012*t^2 - 5232*t^3 - 23092*t^4 - 90160*t^5 - 
     257184*t^6 - 565888*t^7 - 496704*t^8 - 193536*t^9 - 688128*t^10 + 
     737280*t^11), {0}}}, OreAlgebraObject[{Der[t]}, Together, 
  Together[#1 + #2] & , Together[#1*#2] & , None], DegreeLexicographic]
