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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.

%I A336167 #8 Jul 14 2020 23:16:11
%S A336167 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,
%T A336167 0,111,163,53,2,0,236,342,126,6,0,562,738,280,21,0,1302,1662,634,60,0,
%U A336167 3172,3838,1423,165,1,0,7746,9041,3308,412,7,0,19347,21812,7676,1044,26
%N 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.
%F A336167 O.g.f.: Product_{n>=1} (y/(1 - x^n) - y + 1)^A005195(n).
%e A336167 1,
%e A336167 0, 1,
%e A336167 0, 2,
%e A336167 0, 2,   1,
%e A336167 0, 4,   2,
%e A336167 0, 4,   6,
%e A336167 0, 9,   10,  1,
%e A336167 0, 12,  22,  3,
%e A336167 0, 27,  40,  9,
%e A336167 0, 49,  80,  24,
%e A336167 0, 111, 163, 53,  2.
%t A336167 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];
%t A336167 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
%Y A336167 Cf. A035054 (column k=1), A005195 (row sums).
%K A336167 nonn,tabf
%O A336167 0,5
%A A336167 _Geoffrey Critzer_, Jul 10 2020