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The following Mathematica program:

(*program start*)
(*coefficients (coeff) in power series can be \
changed*)coeff = {1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
nn = Length[coeff];
A1 = Table[Table[If[n >= k, 1, 0], {k, 1, nn}], {n, 1, nn}];
MatrixForm[A1];
(*Iteratively downshift the matrix A1 one step and apply power series \
with coefficients of choice (coeff)*)
Monitor[Do[A2 = A1[[1 ;; nn - 1]];
  A2 = Prepend[A2, ConstantArray[0, nn]];
  A1 = Total[
    Table[Table[
      Table[(coeff)[[m]]*MatrixPower[A2, m - 1][[n, k]], {k, 1, 
        nn}], {n, 1, nn}], {m, 1, nn}]];, {j, 1, nn + 1}], j]
MatrixForm[A1];
A1[[All, 1]]
(*program end*)

gives a signed version of the central binomial coefficients as output:

$1, -1, 2, -3, 6, -10, 20, -35, 70, -126, 252, -462...$

A divisibility recurrence analog of what is a variant of series reversion, is a as follows:

Clear[t, n, k, i, x]
coeff = {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];
  A1 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  A2 = A1[[1 ;; nn - 1]];
  A2 = Prepend[A2, ConstantArray[0, nn]];
cc = Sum[
    coeff[[n]]*MatrixPower[A2, n - 1][[All, 1]], {n, 1, nn}];, {i, 1, 
   nn}], i]
cc

With the same input sequence in the coefficients I then get:

$1, -1, 2, -4, 8, -16, 32, -64, 128, -256, 512, -1024$

which is a signed version of the powers of two, excluding the first term.

Since

(*program start*)(*coefficients (coeff) in power series can be \
changed*)coeff = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1};
nn = Length[coeff];
A1 = Table[Table[If[n >= k, 1, 0], {k, 1, nn}], {n, 1, nn}];
MatrixForm[A1];
(*Iteratively downshift the matrix A1 one step and apply power series \
with coefficients of choice (coeff)*)
Monitor[Do[A2 = A1[[1 ;; nn - 1]];
  A2 = Prepend[A2, ConstantArray[0, nn]];
  A1 = Total[
    Table[Table[
      Table[(coeff)[[m]]*MatrixPower[A2, m - 1][[n, k]], {k, 1, 
        nn}], {n, 1, nn}], {m, 1, nn}]];, {j, 1, nn + 1}], j]
MatrixForm[A1];
A1[[All, 1]]
(*program end*)

gives the Catalan numbers:

$1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786,...$

is there a closed form formula for the output of the so called divisibility recurrence analog of the Catalan numbers?

Clear[t, n, k, i, x]
coeff = {1, 1, 1, 1, 1, 1, 1, 1, 1, 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];
  A1 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  A2 = A1[[1 ;; nn - 1]];
  A2 = Prepend[A2, ConstantArray[0, nn]];
  cc = Sum[
    coeff[[n]]*MatrixPower[A2, n - 1][[All, 1]], {n, 1, nn}];, {i, 1, 
   nn}], i]
cc

That is, the output from this last program:

1, 1, 2, 4, 10, 24, 66, 178, 508, 1464, 4320, 12886, 38992, 119030, 366740, 1138036, 3554962, 11167292, 35259290, 111825840, 356100044, 1138107490, 3649507278, 11738028470

is the sequence I would like to know if it has a closed form.

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