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b[s_] := b[s] = Expand[Function[n, If[n==0, 1, Sum[Function[t, If[t==0, 0, If[t==1, x, 1]*b[s~Complement~{j}]]][Abs[n-j]], {j, s}]]][Length[s]]];
T[n_] := PadRight[CoefficientList[b[Range[n]], x], n+1];
T /@ Range[0, 12] // Flatten (* Jean-Fran�ois Alcover, Feb 09 2021, after Alois P. Heinz *)
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Sum_{k=1..n} T(n,k) = A296050(n).
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