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

0,5

Comments

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

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,  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, ...
  ...

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(EulerXY(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 A329055.

Antidiagonal sums are A329053.

The equivalent array for labeled nodes is A328887.

Cf. A329054.

Sequence in context: A108086 A130595 A108363 * A076831 A197061 A230861

Adjacent sequences: A329049 A329050 A329051 * A329053 A329054 A329055

Keyword

nonn,tabl

Author

Andrew Howroyd, Nov 02 2019

Status

approved