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COMMENTS
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If the first two rows of A124027 are left off, the wrong answer is given for the number of program representations necessary to be tested. My machine won't calculate to the next one at n=25. This line of reasoning also produces the sequence: Flatten[Table[If[c[[n]] == 1, n, {}], {n, 1, Length[c]}]] {7, 14, 20, 21, 25, 30, 31, 33, 37, 38, 39, 40, 41, 42, 45, 47, 48, 49, 51, 52, 53, 55, 60}
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MATHEMATICA
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(*A079365*); c = {0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0}; (*A124027*); p[0, x] = 0; p[1, x] = x; p[2, x] = 1; p[k_, x_] := p[k, x] = Sum[ p[j, x]*p[k - j, x], {j, 2, k - 1}]; Flatten[Table[If[c[[n + 1]] == 1, Sum[CoefficientList[p[n, x], x][[m]], {m, 1, Length[CoefficientList[p[n, x], x]]}], {}], {n, 0, 21}] ]
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