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.
Original entry on oeis.org
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, 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, 19, 4, 1, 0, 0, 1, 5, 24, 73, 132, 132, 73, 24, 5, 1, 0
Offset: 0
Array begins:
===================================================
n\m | 0 1 2 3 4 5 6 7 8
----+----------------------------------------------
0 | 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2 | 0, 1, 1, 2, 2, 3, 3, 4, 4, ...
3 | 0, 1, 2, 4, 7, 10, 14, 19, 24, ...
4 | 0, 1, 2, 7, 14, 28, 45, 73, 105, ...
5 | 0, 1, 3, 10, 28, 65, 132, 242, 412, ...
6 | 0, 1, 3, 14, 45, 132, 316, 693, 1349, ...
7 | 0, 1, 4, 19, 73, 242, 693, 1742, 3927, ...
8 | 0, 1, 4, 24, 105, 412, 1349, 3927, 10079, ...
...
The equivalent array for labeled nodes is
A072590.
-
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)))}
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(1 + s)}
{ my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
A329052
Array read by antidiagonals: T(n,m) is the number of unlabeled bicolored acyclic graphs with n nodes of one color and m of the other.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 21, 15, 6, 1, 1, 7, 21, 38, 38, 21, 7, 1, 1, 8, 28, 62, 82, 62, 28, 8, 1, 1, 9, 36, 95, 158, 158, 95, 36, 9, 1, 1, 10, 45, 138, 278, 356, 278, 138, 45, 10, 1, 1, 11, 55, 192, 459, 724, 724, 459, 192, 55, 11, 1
Offset: 0
Array begins:
=======================================================
n\m | 0 1 2 3 4 5 6 7 8
----+--------------------------------------------------
0 | 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
2 | 1, 3, 6, 10, 15, 21, 28, 36, 45, ...
3 | 1, 4, 10, 21, 38, 62, 95, 138, 192, ...
4 | 1, 5, 15, 38, 82, 158, 278, 459, 716, ...
5 | 1, 6, 21, 62, 158, 356, 724, 1359, 2388, ...
6 | 1, 7, 28, 95, 278, 724, 1690, 3612, 7143, ...
7 | 1, 8, 36, 138, 459, 1359, 3612, 8731, 19404, ...
8 | 1, 9, 45, 192, 716, 2388, 7143, 19404, 48213, ...
...
The equivalent array for labeled nodes is
A328887.
-
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)))}
R(n)={my(A=O(x)); for(j=1, 2*n, A = if(j%2, 1, y)*x*EulerXY(A)); A};
P(n)={my(r1=R(n), r2=x*EulerXY(r1), s=r1+r2-r1*r2); Vec(EulerXY(s))}
{ my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }
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