A048159 Triangle giving a(n,k) = number of (n,k) labeled Greg trees (n >= 2, 0 <= k <= n-2).

1, 3, 1, 16, 13, 3, 125, 171, 85, 15, 1296, 2551, 2005, 735, 105, 16807, 43653, 47586, 26950, 7875, 945, 262144, 850809, 1195383, 924238, 412650, 100485, 10395, 4782969, 18689527, 32291463, 31818045, 19235755, 7113645, 1486485, 135135

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• tabl
• nice

Comments

An (n,k) Greg tree can be described as a tree with n black nodes and k white nodes where only the black nodes are labeled and the white nodes are of degree at least 3.

Row sums give A005263.

Examples

			Triangle begins
    1;
    3,   1;
   16,  13,   3;
  125, 171,  85,  15;
  ...
		

Links

• C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128.
• C. Flight, How many stemmata?, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)
• C. Flight, Letter to N. J. A. Sloane, Nov 1990
• M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
• Lucas Randazzo, Arboretum for a generalization of Ramanujan polynomials, arXiv:1905.02083 [math.CO], 2019.
• Index entries for sequences related to trees

Crossrefs

Cf. A005264, A048160, A052300, A052301, A052302, A052303.

Programs

Mathematica

a[n_, 0] := n^(n-2); a[n_ /; n >= 2, k_] /; 0 <= k <= n-2 := a[n, k] = (n+k-3)*a[n-1, k-1] + (2*n+2*k-3)*a[n-1, k] + (k+1)*a[n-1, k+1]; a[n_, k_] = 0; Table[a[n, k], {n, 2, 9}, {k, 0, n-2}] // Flatten (* Jean-François Alcover, Oct 03 2013 *)

Formula

a(n, 0) = n^(n-2), a(n, k) = (n+k-3)*a(n-1, k-1) + (2n+2k-3)*a(n-1, k) + (k+1)*a(n-1, k+1).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000