A336096 Irregular triangular array read by rows. T(n,k) is the number of unlabeled forests of distinct trees on n nodes containing exactly k trees.
1, 1, 1, 1, 2, 1, 3, 3, 6, 5, 1, 11, 11, 2, 23, 21, 5, 47, 46, 12, 106, 96, 27, 2, 235, 216, 62, 4, 551, 482, 142, 13, 1301, 1121, 328, 33, 3159, 2633, 763, 87, 1, 7741, 6334, 1809, 211, 6, 19320, 15414, 4322, 532, 18, 48629, 38132, 10488, 1301, 55, 123867, 95321, 25710, 3232, 157, 317955, 241029, 63802, 7996, 429, 3, 823065, 614862, 159817, 19973, 1149, 12
Offset: 1
Examples
Triangle begins:
1;
1;
1, 1;
2, 1;
3, 3;
6, 5, 1;
11, 11, 2;
23, 21, 5;
47, 46, 12;
106, 96, 27, 2;
...
Programs
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Mathematica
nn = 20; 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]; Map[Select[#, # > 0 &] &, Drop[CoefficientList[ Series[Product[(1 + y x^n)^b[[n]], {n, 1, nn}], {x, 0, nn}], {x,y}], 1]] // Grid
Formula
O.g.f.: Product_n>=1 (1+ y*x^n)^A000055(n).