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A055541
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Total number of leaves (nodes of vertex degree 1) in all labeled trees with n nodes.
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9
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0, 2, 6, 36, 320, 3750, 54432, 941192, 18874368, 430467210, 11000000000, 311249095212, 9659108818944, 326173191714734, 11905721598812160, 467086816406250000, 19599665578316398592, 875901453762003632658, 41532319635035234107392, 2082547005958224830656820
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OFFSET
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1,2
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COMMENTS
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Equivalently, a(n) is the number of rooted labeled trees such that the root node has degree 1. - Geoffrey Critzer, Feb 07 2012
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Tree Leaf.
Index entries for sequences related to trees
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FORMULA
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From Vladeta Jovovic, Mar 31 2001: (Start)
a(n) = n*(n-1)^(n-2), n > 1.
E.g.f.: -x*LambertW(-x). (End)
a(n) = Sum_{k=1..n} (A055314(n, k)*k). - Christian G. Bower, Jun 12 2000
E.g.f.: x*T(x) where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Feb 07 2012
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A003227, A003228, A055314, A055540, A055897.
Essentially the same as A061302.
Sequence in context: A182037 A306066 A061302 * A275551 A321085 A133822
Adjacent sequences: A055538 A055539 A055540 * A055542 A055543 A055544
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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EXTENSIONS
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More terms from Christian G. Bower, Jun 12 2000
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STATUS
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approved
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