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%I #10 Apr 05 2023 15:35:47
%S 1,1,1,1,1,1,2,1,2,3,1,3,7,1,3,10,9,1,4,14,28,1,4,19,45,37,1,5,24,73,
%T 132,1,5,30,105,242,168,1,6,37,152,412,693,1,6,44,204,660,1349,895,1,
%U 7,52,274,1008,2472,3927,1,7,61,351,1479,4219,8105,5097,1,8
%N Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= floor(n/2), except k = 0 if n = 1 ) nodes of one color and n-k nodes of the other color (the colors are interchangeable).
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
%H R. W. Robinson, <a href="/A122087/b122087.txt">Rows 1 through 30, flattened</a>
%F T(n,k) = A329054(k, n-k) for 2*k < n; T(2*n,n) = A119856(n). - _Andrew Howroyd_, Apr 04 2023
%e K M N gives the number N of unlabeled free bicolored trees with K nodes of one color and M nodes of the other color.
%e 0 1 1
%e Total( 1) = 1
%e 1 1 1
%e Total( 2) = 1
%e 1 2 1
%e Total( 3) = 1
%e 1 3 1
%e 2 2 1
%e Total( 4) = 2
%e 1 4 1
%e 2 3 2
%e Total( 5) = 3
%e 1 5 1
%e 2 4 2
%e 3 3 3
%e Total( 6) = 6
%e 1 6 1
%e 2 5 3
%e 3 4 7
%e Total( 7) = 11
%e 1 7 1
%e 2 6 3
%e 3 5 10
%e 4 4 9
%e Total( 8) = 23
%e From _Andrew Howroyd_, Apr 05 2023: (Start)
%e Triangle begins:
%e n\k| 0 1 2 3 4 5 6
%e ----+----------------------------
%e 1 | 1;
%e 2 | . 1;
%e 3 | . 1;
%e 4 | . 1, 1;
%e 5 | . 1, 2;
%e 6 | . 1, 2, 3;
%e 7 | . 1, 3, 7;
%e 8 | . 1, 3, 10, 9;
%e 9 | . 1, 4, 14, 28;
%e 10 | . 1, 4, 19, 45, 37;
%e 11 | . 1, 5, 24, 73, 132;
%e 12 | . 1, 5, 30, 105, 242, 168;
%e ...
%e (End)
%Y Row sums give A000055.
%Y Cf. A119856, A329054, A362019 (labeled version).
%K nonn,tabf
%O 1,7
%A _N. J. A. Sloane_, Oct 19 2006
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