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A055541 Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.

%I A055541 #41 Feb 16 2025 08:32:43
%S A055541 0,2,6,36,320,3750,54432,941192,18874368,430467210,11000000000,
%T A055541 311249095212,9659108818944,326173191714734,11905721598812160,
%U A055541 467086816406250000,19599665578316398592,875901453762003632658,41532319635035234107392,2082547005958224830656820
%N A055541 Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.
%C A055541 Equivalently, a(n) is the number of rooted labeled trees such that the root node has degree 1. - _Geoffrey Critzer_, Feb 07 2012
%H A055541 G. C. Greubel, <a href="/A055541/b055541.txt">Table of n, a(n) for n = 1..385</a>
%H A055541 Dixy Msapato, <a href="https://arxiv.org/abs/2002.12194">Counting the number of tau-exceptional sequences over Nakayama algebras</a>, arXiv:2002.12194 [math.RT], 2020.
%H A055541 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TreeLeaf.html">Tree Leaf.</a>
%H A055541 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A055541 From _Vladeta Jovovic_, Mar 31 2001: (Start)
%F A055541 a(n) = n*(n-1)^(n-2), n > 1.
%F A055541 E.g.f.: -x*LambertW(-x). (End)
%F A055541 a(n) = Sum_{k=1..n} (A055314(n, k)*k). - _Christian G. Bower_, Jun 12 2000
%F A055541 E.g.f.: x*T(x) where T(x) is the e.g.f. for A000169. - _Geoffrey Critzer_, Feb 07 2012
%t A055541 Join[{0,2}, Table[Sum[n!/k! StirlingS2[n-2,n-k] k, {k,2,n-1}], {n,3,20}]] (* _Geoffrey Critzer_, Nov 22 2011 *)
%t A055541 Join[{0,2}, Table[n*(n-1)^(n-2), {n,3,50}]] (* or *) Rest[With[{nmax = 40}, CoefficientList[Series[-x*LambertW[-x], {x, 0, nmax}], x]*Range[0, nmax]!]] (* _G. C. Greubel_, Nov 11 2017 *)
%o A055541 (PARI) for(n=1, 30, print1(if(n==1, 0, if(n==2, 2, n*(n-1)^(n-2))), ", ")) \\ _G. C. Greubel_, Nov 11 2017
%o A055541 (Magma) [0,2] cat [n*(n-1)^(n-2): n in [3..10]]; // _G. C. Greubel_, Nov 11 2017
%Y A055541 Cf. A003227, A003228, A055314, A055540, A055897.
%Y A055541 Essentially the same as A061302.
%K A055541 nonn
%O A055541 1,2
%A A055541 _Eric W. Weisstein_
%E A055541 More terms from _Christian G. Bower_, Jun 12 2000