A206855 The sum of the degree of each root node over all rooted labeled trees on n nodes.

0, 0, 2, 12, 96, 1000, 12960, 201684, 3670016, 76527504, 1800000000, 47158953820, 1362182012928, 43011849456888, 1474041721757696, 54493461914062500, 2161727821137838080, 91597537648314105376, 4128944057284204560384, 197293926880252878693804, 9961472000000000000000000

Offset: 0

Author

Geoffrey Critzer, Feb 13 2012

Keywords

• nonn
• easy

Comments

The mean root degree approaches 2 as n -> infinity.

Links

• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 179.

Programs

Mathematica

nn=15;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];D[ Range[0,nn]!CoefficientList[Series[x Exp[y t],{x,0,nn}],x],y]/.y->1

Formula

a(n) = Sum_{k=0..n} A206429(n,k)*k.

E.g.f.: T(x)^2 where T(x) is the e.g.f. for A000169.

a(n) = 2*(n^(n-1) - n^(n-2)).

a(n) = 2*A053506(n). - R. J. Mathar, Nov 07 2014

Extensions

a(10) corrected by Georg Fischer, Mar 23 2023