A336167 Irregular triangular array read by rows. T(n,k) is the number of forests on n unlabeled nodes with exactly k distinct isomorphism classes of trees.
1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 2, 0, 4, 6, 0, 9, 10, 1, 0, 12, 22, 3, 0, 27, 40, 9, 0, 49, 80, 24, 0, 111, 163, 53, 2, 0, 236, 342, 126, 6, 0, 562, 738, 280, 21, 0, 1302, 1662, 634, 60, 0, 3172, 3838, 1423, 165, 1, 0, 7746, 9041, 3308, 412, 7, 0, 19347, 21812, 7676, 1044, 26
Offset: 0
Examples
1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 2, 0, 4, 6, 0, 9, 10, 1, 0, 12, 22, 3, 0, 27, 40, 9, 0, 49, 80, 24, 0, 111, 163, 53, 2.
Programs
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Mathematica
nn = 25; f[x_] := Sum[a[n] x^n, {n, 0, nn}]; sol = SolveAlways[0 == Series[ f[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; r[x_] := Sum[a[n] x^n, {n, 0, nn}] /. sol; b = Drop[Flatten[CoefficientList[Series[r[x] - 1/2 (r[x]^2 - r[x^2]), {x, 0, nn}], x]], 1]; h[list_] := Prepend[Select[list, # > 0 &], 0]; Prepend[Drop[Map[h, CoefficientList[Series[Product[(y/(1 - x^k) - y + 1)^b[[k]], {k, 1, nn}], {x, 0, nn}], {x, y}]], 1], {1}] // Grid
Formula
O.g.f.: Product_{n>=1} (y/(1 - x^n) - y + 1)^A005195(n).