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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n < 1, 1, Expand[Sum[ If[j < i, x, 1] *b[n - 1, j, t + If[j == i, 1, 0]], {j, 0, t + 1}]]]; a[n_] := Coefficient[b[n - 1, 0, 0], x, 7]; Table[a[n], {n, 13, 40}] (* Jean-François Alcover, Feb 09 2015, after A242352 *)
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