A071211 Triangular array T(n,k) read by rows, giving number of labeled free trees such that the root is smaller than all its children, with respect to the number n of vertices and to the label k of the root.

1, 3, 1, 16, 8, 3, 125, 75, 40, 16, 1296, 864, 540, 300, 125, 16807, 12005, 8232, 5292, 3024, 1296, 262144, 196608, 143360, 100352, 65856, 38416, 16807, 4782969, 3720087, 2834352, 2099520, 1492992, 995328, 589824, 262144, 100000000

Offset: 1

Author

Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002

Keywords

• easy
• nonn
• tabl

References

• C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.

Crossrefs

Cf. A000312, A000272 (first column).

Programs

Maple

T:= (n, k)-> (n-k)*(n+1)^(n-k-1)*n^(k-1):
seq(seq(T(n, k), k=0..n-1), n=1..10);

PARI

tabl(nn) = {for (n=1, nn, for (k=0, n-1, print1((n-k)*(n+1)^(n-k-1)*n^(k-1), ", ");); print(););} \\ Michel Marcus, Jun 27 2013

Formula

T(n,k) = (n-k)*(n+1)^(n-k-1)*n^(k-1).