A036276 a(n) = A001864(n+1)/2.

0, 1, 12, 156, 2360, 41400, 831012, 18832576, 476200944, 13301078400, 406907517500, 13534968927744, 486470108273448, 18790567023993856, 776343673316956500, 34165751933338828800, 1595693034061797583328, 78831769938218360930304, 4107393289066148637198444

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

Cf. A001864, A264902.

Sequence in context: A158546 A110216 A218839 * A003130 A015000 A220225

Adjacent sequences: A036273 A036274 A036275 * A036277 A036278 A036279

Keyword

nonn,easy

Author

N. J. A. Sloane.

Status

approved