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A329054 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.

%I A329054 #9 Jan 09 2020 19:26:14
%S A329054 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 A329054 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 A329054 19,4,1,0,0,1,5,24,73,132,132,73,24,5,1,0
%N A329054 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 A329054 The two color classes are not interchangeable. Adjacent nodes cannot have the same color.
%C A329054 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 A329054 Andrew Howroyd, <a href="/A329054/b329054.txt">Table of n, a(n) for n = 0..1325</a>
%e A329054 Array begins:
%e A329054 ===================================================
%e A329054 n\m | 0  1  2   3    4    5     6     7      8
%e A329054 ----+----------------------------------------------
%e A329054   0 | 1, 1, 0,  0,   0,   0,    0,    0,     0, ...
%e A329054   1 | 1, 1, 1,  1,   1,   1,    1,    1,     1, ...
%e A329054   2 | 0, 1, 1,  2,   2,   3,    3,    4,     4, ...
%e A329054   3 | 0, 1, 2,  4,   7,  10,   14,   19,    24, ...
%e A329054   4 | 0, 1, 2,  7,  14,  28,   45,   73,   105, ...
%e A329054   5 | 0, 1, 3, 10,  28,  65,  132,  242,   412, ...
%e A329054   6 | 0, 1, 3, 14,  45, 132,  316,  693,  1349, ...
%e A329054   7 | 0, 1, 4, 19,  73, 242,  693, 1742,  3927, ...
%e A329054   8 | 0, 1, 4, 24, 105, 412, 1349, 3927, 10079, ...
%e A329054   ...
%o A329054 (PARI)
%o A329054 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 A329054 R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
%o A329054 P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(1 + s)}
%o A329054 { my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
%Y A329054 Main diagonal is A119857.
%Y A329054 Antidiagonal sums are A122086.
%Y A329054 The equivalent array for labeled nodes is A072590.
%Y A329054 Cf. A122085, A329053.
%K A329054 nonn,tabl
%O A329054 0,18
%A A329054 _Andrew Howroyd_, Nov 02 2019