A129137 Number of trees on [n], rooted at 1, in which 2 is a descendant of 3.

0, 0, 1, 5, 37, 366, 4553, 68408, 1206405, 24447440, 560041201, 14315792256, 404057805989, 12482986261760, 419042630871225, 15189786100468736, 591374264243364037, 24612549706061862912, 1090556290466098198625

Offset

1,4

Links

Washington G. Bomfim, Table of n, a(n) for n = 1..50

H. Bergeron, E. M. F. Curado, J. P. Gazeau and L. M. C. S. Rodrigues, A note about combinatorial sequences and Incomplete Gamma function, arXiv preprint arXiv: 1309.6910, 2013

Formula

The following formula counts these trees by the length r of the path from 1 to 3: Sum_{r=1..n-2} (n-3)!*n^(n-2-r)/(n-2-r)!.

Example

a(4)=5 counts {1->3->2, 1->4}, {1->3->2, 3->4}, {1->3->2->4}, {1->3->4->2}, {1->4->3->2}.

Mathematica

Table[Exp[n]*Gamma[n-2, n] // Round, {n, 1, 50}] (* Jean-François Alcover, Jan 15 2014 *)

Crossrefs

Cf. A057500 = binomial(n-1, 2)*a(n).

Sequence in context: A025168 A084358 A050351 * A357397 A276232 A055869

Adjacent sequences: A129134 A129135 A129136 * A129138 A129139 A129140

Keyword

nonn

Author

David Callan, Mar 30 2007

Status

approved