A329910 - OEIS

%I #22 Nov 24 2019 04:13:01

%S 0,0,1,4,32,72,2187,20736,262144,3200000,48828125,729000000,

%T 13060694016,230539333248,4747561509943,96717311574016,

%U 2251799813685250,51998697814229000,1350851717672990000,34867844010000000000,1000000000000000000000,28531167061100000000000

%N Number of harmoniously labeled graphs with n edges and at most n vertices.

%C A graph G with n edges is harmonious if there is an injection f from its vertex set to the group of integers modulo n such that when each edge uv of G is assigned the weight f(u)+f(v) (mod n), the resulting weights are distinct.

%H R. L. Graham and N. J. A. Sloane, <a href="http://dx.doi.org/10.1137/0601045">On Additive Bases and Harmonious Graphs</a>, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.

%H R. L. Graham and N. J. A. Sloane, <a href="http://neilsloane.com/doc/RLG/073.pdf">On Additive Bases and Harmonious Graphs</a>

%F For n odd, a(n) = ((n-1)/2)^n. For n even, a(n) = (n*(n-2)/4)^(n/2).

%e a(3)=1 because there is only one harmonious graph with 3 edges and at most 3 vertices.

%t Table[If[EvenQ[n],(n*(n-2)/4)^(n/2),((n-1)/2)^n],{n,1,22}] (* _Stefano Spezia_, Nov 24 2019 *)

%Y A085526 contains the odd-indexed terms.

%K nonn,easy

%O 1,4

%A _Christian Barrientos_, Nov 23 2019