%I #32 May 07 2019 07:36:45

%S 1,3,1,16,13,3,125,171,85,15,1296,2551,2005,735,105,16807,43653,47586,

%T 26950,7875,945,262144,850809,1195383,924238,412650,100485,10395,

%U 4782969,18689527,32291463,31818045,19235755,7113645,1486485,135135

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

%C 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.

%C Row sums give A005263.

%H C. Flight, <a href="https://doi.org/10.1484/J.MSS.3.1335">How many stemmata?</a>, Manuscripta, 34 (1990), 122-128.

%H C. Flight, <a href="/A048159/a048159.pdf">How many stemmata?</a>, Manuscripta, 34 (1990), 122-128. (Annotated scanned copy)

%H C. Flight, <a href="/A005263/a005263.pdf">Letter to N. J. A. Sloane, Nov 1990</a>

%H M. Josuat-Vergès, <a href="http://arxiv.org/abs/1310.7531">Derivatives of the tree function</a>, arXiv preprint arXiv:1310.7531 [math.CO], 2013.

%H Lucas Randazzo, <a href="https://arxiv.org/abs/1905.02083">Arboretum for a generalization of Ramanujan polynomials</a>, arXiv:1905.02083 [math.CO], 2019.

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F 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).

%e Triangle begins

%e 1;

%e 3, 1;

%e 16, 13, 3;

%e 125, 171, 85, 15;

%e ...

%t 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 *)

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

%K nonn,easy,tabl,nice

%O 2,2

%A _N. J. A. Sloane_

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