I asked SymPy to try your approach, but I can only see patterns in the first and the last polynomial coefficients.
I am posting the results to let smarter people find those patterns:
−x2+22x3−6x2+6−3x4+20x3−35x2+244x5−45x4+170x3−225x2+120−5x6+84x5−525x4+1470x3−1624x2+7206x7−140x6+1288x5−5880x4+13538x3−13132x2+5040−7x8+216x7−2730x6+18144x5−67347x4+134568x3−118124x2+403208x9−315x8+5220x7−47250x6+253092x5−807975x4+1447360x3−1172700x2+362880