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.

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

Author

Andrew Howroyd, Nov 02 2019

Keywords

• nonn
• tabl

Comments

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

Examples

			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, ...
  ...
		

Links

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

Crossrefs

Main diagonal is A329055.

Antidiagonal sums are A329053.

The equivalent array for labeled nodes is A328887.

Cf. A329054.

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(EulerXY(s))}
{ my(A=P(10)); for(n=0, #A\2, for(k=0, #A\2, print1(polcoef(A[n+k+1], k), ", ")); print) }