Convolution with divisor recurrence giving powers of 2

Clear[t, n, k, i, nn, x]
coeff = {1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
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]]*MatrixPower[A5, n - 1][[All, 1]], {n, 1, nn}]];
  , {i, 1, nn}], i]
cc
CoefficientList[Series[(1 + x)/(1 + 2*x), {x, 0, nn - 1}], x]
Flatten[{1, Table[(-(-2)^(n - 1)), {n, 1, 28 - 1}]}]