A336309 Triangular array read by rows. T(n,k) is the number of labeled rooted unordered binary trees (as in A036774) with path length exactly k, n >= 1, 0 <= k <= C(n,2).

1, 0, 2, 0, 0, 3, 6, 0, 0, 0, 0, 24, 12, 24, 0, 0, 0, 0, 0, 0, 120, 120, 120, 60, 120, 0, 0, 0, 0, 0, 0, 0, 0, 360, 1440, 360, 1440, 720, 720, 360, 720, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 630, 7560, 10080, 10080, 7560, 12600, 7560, 10080, 5040, 5040, 2520, 5040

Offset: 1

Author

Geoffrey Critzer, Jul 17 2020

Keywords

• nonn
• tabf

Comments

The path length of a tree is the distance from the root to a node summed over all nodes in the tree.

Examples

			  1,
  0, 2,
  0, 0, 3, 6,
  0, 0, 0, 0, 24, 12,  24,
  0, 0, 0, 0,  0,  0, 120, 120, 120,   60,   120,
  0, 0, 0, 0,  0,  0,   0,   0, 360, 1440,   360, 1440, 720, 720, 360, 720
		

Crossrefs

Cf. A036774.

Programs

Mathematica

nn = 6; f[z_, u_] := Sum[Sum[a[n, k] u^k z^n/n!, {k, 0, Binomial[n, 2]}], {n, 1,
   nn}]; sol =SolveAlways[Series[0 == f[z, u] - z (1 + f[u z, u] + f[u z, u]^2/2!), {z, 0, nn}], {z, u}];Level[Table[Table[a[n, k], {k, 0, Binomial[n, 2]}], {n, 1, nn}] /. sol, {2}] // Grid

Formula

E.g.f. satisfies A(x,y) = x + x*A(y*x,y) + x*A(y*x,y)^2/2.