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.

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

Author

Andrew Howroyd, Nov 02 2019

Keywords

• nonn
• tabl

Comments

The two color classes are not interchangeable. Adjacent nodes cannot have the same color.

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.

Examples

			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, ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

Crossrefs

Main diagonal is A119857.

Antidiagonal sums are A122086.

The equivalent array for labeled nodes is A072590.

Cf. A122085, A329053.

Programs

PARI

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) }