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

0,18

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.

Links

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

Example

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, ...
  ...

Prog

(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) }

Crossrefs

Main diagonal is A119857.

Antidiagonal sums are A122086.

The equivalent array for labeled nodes is A072590.

Cf. A122085, A329053.

Sequence in context: A368122 A281459 A163528 * A239509 A360985 A335833

Adjacent sequences: A329051 A329052 A329053 * A329055 A329056 A329057

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 02 2019

Status

approved