OFFSET
0,3
COMMENTS
This is Sum_{all n^(n-2) labeled trees T on n nodes} Sum_{1<=i<j<=n} distance(node i, node j).
a(n) is the total number of all defects in defective parking functions of length n+1. - Alois P. Heinz, Nov 28 2015
With offset 1, a(n) is the number of unordered pairs {f,g} where for some nonempty proper subset S of [n], f:S->S and g:[n]\S->[n]\S. - Geoffrey Critzer, Apr 23 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..380
J. Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
Peter Winkler, Mean distance in a tree, in Computational algorithms, operations research and computer science (Burnaby, BC, 1987). Discrete Appl. Math. 27 (1990), no. 1-2, 179-185. [For background information only.]
FORMULA
a(n) = Sum_{k>0} k * A264902(n+1,k). - Alois P. Heinz, Nov 28 2015
MATHEMATICA
Table[Sum[Binomial[n, k] (n - k)^(n - k) k^k, {k, n - 1}]/2, {n, 18}] (* Michael De Vlieger, Apr 24 2017, after Harvey P. Dale at A001864 *)
PROG
(Python)
from math import math
def A036276(n): return sum(comb(n+1, k)*(n+1-k)**(n+1-k)*k**k for k in range(1, (n>>1)+1)) + (comb(n+1, m:=n+1>>1)*m**(n+1)>>1 if n&1 else 0) # Chai Wah Wu, Apr 26 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved