Mathematica 8:

Clear[nn, X, A]

nn = 12;

A = Range[nn + 1]*0;

X = Transpose[Table[Table[Binomial[n, k], {n, 0, nn}], {k, 0, nn}]] -

IdentityMatrix[nn + 1];

Do[A[[i + 1]] = MatrixPower[X, i][[All, 1]], {i, 0, nn}]

MatrixForm[Transpose[A]]

Total[A]

Matrix A is triangle A131689 in the oeis:

A131689

Which outputs:

{1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632573, 28091567595}

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