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A122087 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).

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