A369569 a(n) = (n-1)! * n^n.

1, 4, 54, 1536, 75000, 5598720, 592950960, 84557168640, 15620794116480, 3628800000000000, 1035338990313196800, 355902198372945100800, 145077660657859734604800, 69194697632491737238732800, 38174841090323437500000000000, 24122334398245883325016178688000

Offset: 1

Author

Thomas Scheuerle, Jan 26 2024

Keywords

• nonn

Comments

The number of ways n different tags can be assigned to different nodes of an unspecified labeled rooted tree with n nodes. (This therefore includes the choice of one of the n^(n-1) labeled rooted trees.) In this description, we differentiate between labels and tags: we view the labels together with the root as part of the labeled rooted tree's definition, but the tags as an assignment in relation to the labels that is independent of the root.

Is this, equivalently, the number of doubly labeled rooted trees?

Examples

			The 4 labeled rooted trees with two nodes and two tags assigned are:
.
  R        R
  L1--L2   L1--L2
  T1  T2   T2  T1
.
       R        R
  L1--L2   L1--L2
  T1  T2   T2  T1
.
		

Crossrefs

Cf. A000312, A000169, A061711.

Programs

Maple

seq(n^n*factorial(n-1), n=1..16)

Mathematica

Table[n^n*(n-1)!, {n, 1, 16}]

PARI

a(n) = (n-1)!*n^n

Formula

a(n) = n! * n^(n-1).

a(n) = Integral_{x>=0} x^(n-1) * exp(-x/n) dx.

a(n) = n! * [x^n] (1/n)*sinh(n*x)^n. - Stefano Spezia, Feb 21 2024