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%I #53 Feb 24 2024 11:08:44
%S 1,4,54,1536,75000,5598720,592950960,84557168640,15620794116480,
%T 3628800000000000,1035338990313196800,355902198372945100800,
%U 145077660657859734604800,69194697632491737238732800,38174841090323437500000000000,24122334398245883325016178688000
%N a(n) = (n-1)! * n^n.
%C 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 Is this, equivalently, the number of doubly labeled rooted trees?
%F a(n) = n! * n^(n-1).
%F a(n) = Integral_{x>=0} x^(n-1) * exp(-x/n) dx.
%F a(n) = n! * [x^n] (1/n)*sinh(n*x)^n. - _Stefano Spezia_, Feb 21 2024
%e The 4 labeled rooted trees with two nodes and two tags assigned are:
%e .
%e R R
%e L1--L2 L1--L2
%e T1 T2 T2 T1
%e .
%e R R
%e L1--L2 L1--L2
%e T1 T2 T2 T1
%e .
%p seq(n^n*factorial(n-1), n=1..16)
%t Table[n^n*(n-1)!, {n, 1, 16}]
%o (PARI) a(n) = (n-1)!*n^n
%Y Cf. A000312, A000169, A061711.
%K nonn
%O 1,2
%A _Thomas Scheuerle_, Jan 26 2024
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