A122085 -id:A122085 - cp's OEIS frontend

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

Andrew Howroyd, Nov 02 2019

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.

			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, ...
  ...

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

Main diagonal is A119857.
Antidiagonal sums are A122086.
The equivalent array for labeled nodes is A072590.
Cf. A122085, A329053.

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) }

A122086 Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).

Original entry on oeis.org

2, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793

Offset: 1

N. J. A. Sloane, Oct 19 2006

nonn

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row sums of A122085.
Antidiagonal sums of A329054.
Same as A125702 except for n = 1.
Cf. A000055, A000081.

\\ here TreeGf is A000081 as g.f.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2)} \\ Andrew Howroyd, Nov 02 2019

For n even, a(n) = 2*A000055(n) - A000081(n/2), for n odd, a(n) = 2*A000055(n).

G.f.: 2*f(x) - f(x)^2 where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019

Edited by Christian G. Bower and Franklin T. Adams-Watters, Jan 05 2007

cp's OEIS Frontend

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.

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