OrePolynomial[{{-3*t^3 - 2*t^4 + 38*t^5 + 84*t^6 + 352*t^7 - 128*t^8 - 
    3520*t^9 - 4096*t^10 - 15360*t^11 - 29184*t^12 + 32256*t^13 - 
    82944*t^14 + 110592*t^15, {5}}, 
  {2*(-21*t^2 + 20*t^3 + 490*t^4 + 732*t^5 + 2132*t^6 - 6752*t^7 - 
     46496*t^8 - 74752*t^9 - 202368*t^10 - 310272*t^11 + 294912*t^12 - 
     857088*t^13 + 1188864*t^14), {4}}, 
  {4*(-39*t + 158*t^2 + 1693*t^3 + 932*t^4 - 310*t^5 - 45972*t^6 - 
     197568*t^7 - 371520*t^8 - 833184*t^9 - 1021248*t^10 + 774144*t^11 - 
     2737152*t^12 + 3981312*t^13), {3}}, 
  {24*(-6 + 78*t + 597*t^2 - 748*t^3 - 4086*t^4 - 31260*t^5 - 103084*t^6 - 
     212128*t^7 - 411648*t^8 - 402432*t^9 + 203328*t^10 - 1036800*t^11 + 
     1603584*t^12), {2}}, {24*(48 + 279*t - 2032*t^2 - 7158*t^3 - 37940*t^4 - 
     103600*t^5 - 229568*t^6 - 381984*t^7 - 298560*t^8 + 43776*t^9 - 
     718848*t^10 + 1216512*t^11), {1}}, 
  {96*(-3 - 162*t - 450*t^2 - 2132*t^3 - 4902*t^4 - 11952*t^5 - 15984*t^6 - 
     10080*t^7 - 4320*t^8 - 20736*t^9 + 41472*t^10), {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
