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%I #9 Jan 09 2020 19:26:14
%S 1,1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,1,2,2,1,0,0,1,2,4,2,1,0,0,1,3,7,7,3,
%T 1,0,0,1,3,10,14,10,3,1,0,0,1,4,14,28,28,14,4,1,0,0,1,4,19,45,65,45,
%U 19,4,1,0,0,1,5,24,73,132,132,73,24,5,1,0
%N Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored trees with n nodes of one color and m of the other.
%C The two color classes are not interchangeable. Adjacent nodes cannot have the same color.
%C Essentially the same data as given in the irregular triangle A122085, but including complete columns for n = 0 and m = 0 to give a regular array.
%H Andrew Howroyd, <a href="/A329054/b329054.txt">Table of n, a(n) for n = 0..1325</a>
%e Array begins:
%e ===================================================
%e n\m | 0 1 2 3 4 5 6 7 8
%e ----+----------------------------------------------
%e 0 | 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2 | 0, 1, 1, 2, 2, 3, 3, 4, 4, ...
%e 3 | 0, 1, 2, 4, 7, 10, 14, 19, 24, ...
%e 4 | 0, 1, 2, 7, 14, 28, 45, 73, 105, ...
%e 5 | 0, 1, 3, 10, 28, 65, 132, 242, 412, ...
%e 6 | 0, 1, 3, 14, 45, 132, 316, 693, 1349, ...
%e 7 | 0, 1, 4, 19, 73, 242, 693, 1742, 3927, ...
%e 8 | 0, 1, 4, 24, 105, 412, 1349, 3927, 10079, ...
%e ...
%o (PARI)
%o EulerXY(A)={my(j=serprec(A,x)); exp(sum(i=1, j, 1/i * subst(subst(A + x * O(x^(j\i)), x, x^i), y, y^i)))}
%o R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
%o P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(1 + s)}
%o { my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
%Y Main diagonal is A119857.
%Y Antidiagonal sums are A122086.
%Y The equivalent array for labeled nodes is A072590.
%Y Cf. A122085, A329053.
%K nonn,tabl
%O 0,18
%A _Andrew Howroyd_, Nov 02 2019
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