A034854 Triangle giving number of labeled trees with n >= 3 nodes and diameter d >= 2.
3, 4, 12, 5, 60, 60, 6, 210, 720, 360, 7, 630, 6090, 7560, 2520, 8, 1736, 47040, 112560, 80640, 20160, 9, 4536, 363384, 1496880, 1829520, 907200, 181440, 10, 11430, 2913120, 19207440, 36892800, 28274400, 10886400, 1814400
Offset: 0
Examples
Triangle begins: 3; 4, 12; 5, 60, 60; 6, 210, 720, 360; 7, 630, 6090, 7560, 2520; ...
Links
- J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.[broken link]
- J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
- J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
- Index entries for sequences related to trees
Formula
Reference gives recurrence.
From Geoffrey Critzer, Aug 02 2022: (Start)
Let G_k(x) be the e.g.f. counting the number of rooted labeled trees with height <= k. Then G_k(x) is defined recursively by G_0(x) = x, G_k(x) = x*exp(G_{k-1}(x)). Let H_k(x) be the e.g.f. counting rooted labeled trees of height k. Then H_0(x) = x, H_k(x) = G_k(x) - G_{k-1}(x) for k >= 1. The e.g.f. for column d = 2*m+1 is H_m(x)^2/2. The e.g.f. for column d = 2*m is G_{m-1}(x)*(exp(H_{m-1}(x)) - 1 - H_{m-1}(x)). (End)
Extensions
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 27 2004
Name corrected by Geoffrey Critzer, Aug 02 2022
Comments