cp's OEIS Frontend

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.

%I A336096 #31 Feb 19 2025 16:12:42
%S A336096 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,
%T A336096 62,4,551,482,142,13,1301,1121,328,33,3159,2633,763,87,1,7741,6334,
%U A336096 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
%N 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.
%F A336096 O.g.f.: Product_n>=1 (1+ y*x^n)^A000055(n).
%e A336096 Triangle begins:
%e A336096     1;
%e A336096     1;
%e A336096     1,  1;
%e A336096     2,  1;
%e A336096     3,  3;
%e A336096     6,  5,  1;
%e A336096    11, 11,  2;
%e A336096    23, 21,  5;
%e A336096    47, 46, 12;
%e A336096   106, 96, 27, 2;
%e A336096   ...
%t A336096 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[
%t A336096     Series[Product[(1 + y x^n)^b[[n]], {n, 1, nn}], {x, 0, nn}], {x,y}], 1]] // Grid
%Y A336096 Cf. A035055 (row sums), A000055 (column 1), A095133.
%K A336096 nonn,tabf
%O A336096 1,5
%A A336096 _Geoffrey Critzer_, Jul 09 2020