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A106240 Triangle read by rows: T(n,m) = number of unlabeled cographs on n nodes with m connected components.

%I A106240 #17 Dec 24 2015 02:37:45
%S A106240 1,1,1,2,1,1,5,3,1,1,12,7,3,1,1,33,20,8,3,1,1,90,55,22,8,3,1,1,261,
%T A106240 162,63,23,8,3,1,1,766,477,188,65,23,8,3,1,1,2312,1450,564,196,66,23,
%U A106240 8,3,1,1,7068,4446,1732,590,198,66,23,8,3,1,1,21965,13858,5384,1824,598,199,66,23,8,3,1,1
%N A106240 Triangle read by rows: T(n,m) = number of unlabeled cographs on n nodes with m connected components.
%H A106240 Alois P. Heinz, <a href="/A106240/b106240.txt">Rows n = 1..141, flattened</a>
%H A106240 Washington Bomfim, <a href="http://webonfim.vilabol.uol.com.br/A240.html">Illustration of this sequence</a>
%F A106240 T(n, m) = sum over the partitions of n with m parts: 1K1 + 2K2 + ... + nKn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A000669(i)+Ki-1, Ki).
%e A106240 T(10,8) = 3 because the partitions of 10 with 8 parts are 31111111 and 22111111. The partition 31111111 corresponds to 2 graphs and the partition 22111111 corresponds to only one.
%e A106240 T(n,m) = 1, if and only if m>=n-1. Because A000669(1)=A000669(2)=1, the partitions of n with all parts <=2 correspond to summands = 1. If there is only a summand (or partition), the total is equal to 1. It is clear that for m>=n-1 there is only one partition of n with exactly m parts.
%e A106240 Triangle begins:
%e A106240    1,
%e A106240    1,  1,
%e A106240    2,  1,  1,
%e A106240    5,  3,  1, 1,
%e A106240   12,  7,  3, 1, 1,
%e A106240   33, 20,  8, 3, 1, 1,
%e A106240   90, 55, 22, 8, 3, 1, 1,
%Y A106240 Cf. A000669 (first column), A000084 (row sums), A201922.
%K A106240 nonn,tabl
%O A106240 1,4
%A A106240 _Washington Bomfim_, May 06 2005