This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369569 #53 Feb 24 2024 11:08:44
%S A369569 1,4,54,1536,75000,5598720,592950960,84557168640,15620794116480,
%T A369569 3628800000000000,1035338990313196800,355902198372945100800,
%U A369569 145077660657859734604800,69194697632491737238732800,38174841090323437500000000000,24122334398245883325016178688000
%N A369569 a(n) = (n-1)! * n^n.
%C A369569 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.
%C A369569 Is this, equivalently, the number of doubly labeled rooted trees?
%F A369569 a(n) = n! * n^(n-1).
%F A369569 a(n) = Integral_{x>=0} x^(n-1) * exp(-x/n) dx.
%F A369569 a(n) = n! * [x^n] (1/n)*sinh(n*x)^n. - _Stefano Spezia_, Feb 21 2024
%e A369569 The 4 labeled rooted trees with two nodes and two tags assigned are:
%e A369569 .
%e A369569 R R
%e A369569 L1--L2 L1--L2
%e A369569 T1 T2 T2 T1
%e A369569 .
%e A369569 R R
%e A369569 L1--L2 L1--L2
%e A369569 T1 T2 T2 T1
%e A369569 .
%p A369569 seq(n^n*factorial(n-1), n=1..16)
%t A369569 Table[n^n*(n-1)!, {n, 1, 16}]
%o A369569 (PARI) a(n) = (n-1)!*n^n
%Y A369569 Cf. A000312, A000169, A061711.
%K A369569 nonn
%O A369569 1,2
%A A369569 _Thomas Scheuerle_, Jan 26 2024