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

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

Offset

1,2

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?

Links

Table of n, a(n) for n=1..16.

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

Example

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
.

Maple

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000312, A000169, A061711.

Sequence in context: A111161 A216733 A203039 * A201731 A225823 A265004

Adjacent sequences: A369566 A369567 A369568 * A369570 A369571 A369572

Keyword

nonn

Author

Thomas Scheuerle, Jan 26 2024

Status

approved