A055541 - OEIS
A055541
Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.
9
0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658, 41532319635035234107392, 2082547005958224830656820
COMMENTS
Equivalently, a(n) is the number of rooted labeled trees such that the root node has degree 1. - Geoffrey Critzer, Feb 07 2012
FORMULA
a(n) = n*(n-1)^(n-2), n > 1.
E.g.f.: -x*LambertW(-x). (End)
MATHEMATICA
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 *)
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 *)
PROG
(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
(Magma) [0, 2] cat [n*(n-1)^(n-2): n in [3..10]]; // G. C. Greubel, Nov 11 2017