Divisor transform

Clear[t, n, k, i, nn, x];
x = -2; (*x=1 gives this sequence a (n). Try changing the value of \
"x".*)
coeff = Join[{1, x}, (-x)^(Range[20])]
mp[m_, e_] :=
 If[e == 0, IdentityMatrix@Length@m, MatrixPower[m, e]]; nn =
 Length[coeff]; cc = Range[nn]*0 + 1; Monitor[
 Do[Clear[t]; t[n_, 1] := t[n, 1] = cc[[n]];
  t[n_, k_] :=
   t[n, k] =
    If[n >= k,
     Sum[t[n - i, k - 1], {i, 1, k - 1}] -
      Sum[t[n - i, k], {i, 1, k - 1}], 0];
  A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  A5 = A4[[1 ;; nn - 1]]; A5 = Prepend[A5, ConstantArray[0, nn]];
  cc = Total[
    Table[coeff[[n]]*mp[A5, n - 1][[All, 1]], {n, 1, nn}]];, {i, 1,
   nn}], i]; cc
Join[{1}, x*(x - 1)^Range[0, Length[coeff] - 2]]
%% - %