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A122085 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= n-1, except k=0 or 1 if n=1, k=1 if n=2) nodes of one color and n-k nodes of the other color (the colors are not interchangeable).

%I A122085 #11 Nov 03 2019 15:54:54
%S A122085 1,1,1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,7,7,3,1,1,3,10,14,10,3,1,1,4,
%T A122085 14,28,28,14,4,1,1,4,19,45,65,45,19,4,1,1,5,24,73,132,132,73,24,5,1,1,
%U A122085 5,30,105,242,316,242,105,30,5,1,1,6,37,152,412,693,693,412,152
%N A122085 Triangle read by rows: T(n,k) = number of unlabeled free bicolored trees with n nodes (n >= 1) and k (1 <= k <= n-1, except k=0 or 1 if n=1, k=1 if n=2) nodes of one color and n-k nodes of the other color (the colors are not interchangeable).
%D A122085 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.
%H A122085 R. W. Robinson, <a href="/A122085/b122085.txt">Rows 1 through 30, flattened</a>
%e A122085 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 A122085 0 1 1
%e A122085 1 0 1
%e A122085 Total( 1) = 2
%e A122085 1 1 1
%e A122085 Total( 2) = 1
%e A122085 1 2 1
%e A122085 2 1 1
%e A122085 Total( 3) = 2
%e A122085 1 3 1
%e A122085 2 2 1
%e A122085 3 1 1
%e A122085 Total( 4) = 3
%e A122085 1 4 1
%e A122085 2 3 2
%e A122085 3 2 2
%e A122085 4 1 1
%e A122085 Total( 5) = 6
%e A122085 1 5 1
%e A122085 2 4 2
%e A122085 3 3 4
%e A122085 4 2 2
%e A122085 5 1 1
%e A122085 Total( 6) = 10
%e A122085 .
%e A122085 From _Andrew Howroyd_, Nov 02 2019: (Start)
%e A122085 Triangle for n >= 2, 1 <= k < n:
%e A122085    2 | 1;
%e A122085    3 | 1, 1;
%e A122085    4 | 1, 1,  1;
%e A122085    5 | 1, 2,  2,   1;
%e A122085    6 | 1, 2,  4,   2,   1;
%e A122085    7 | 1, 3,  7,   7,   3,   1;
%e A122085    8 | 1, 3, 10,  14,  10,   3,   1;
%e A122085    9 | 1, 4, 14,  28,  28,  14,   4,   1;
%e A122085   10 | 1, 4, 19,  45,  65,  45,  19,   4,  1;
%e A122085   11 | 1, 5, 24,  73, 132, 132,  73,  24,  5, 1;
%e A122085   12 | 1, 5, 30, 105, 242, 316, 242, 105, 30, 5, 1;
%e A122085   ...
%e A122085 (End)
%Y A122085 Row sums give A122086.
%Y A122085 Cf. A329054 (regular array with same data).
%K A122085 nonn,tabf
%O A122085 1,10
%A A122085 _N. J. A. Sloane_, Oct 19 2006