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A340234 Number of harmonious graphs with n edges and at most n vertices, allowing self-loops.

%I A340234 #25 Jan 07 2021 21:07:53
%S A340234 1,2,8,36,243,1728,16384,160000,1953125,24300000,362797056,5489031744,
%T A340234 96889010407,1727094849536,35184372088832,722204136308736,
%U A340234 16677181699666569,387420489000000000,10000000000000000000,259374246010000000000
%N A340234 Number of harmonious graphs with n edges and at most n vertices, allowing self-loops.
%C A340234 A graph G = (V,E) is harmonious if there exists an injective function f_V : V -> {0,1,...,n-1} such that a bijection occurs in the function f_E : E -> {0,...,n-1} after the harmoniously induced edge labels, f_E(v_iv_j) = (f_V(v_i) +f_V(v_j))(mod n), are applied.
%C A340234 A329910 contains the same data for simple graphs.
%H A340234 Joseph A. Gallian, <a href="https://doi.org/10.37236/27">A dynamic survey of graph labeling</a>, Elec. J. Combin., (2014), #DS6.
%H A340234 D. Tanna, <a href="https://www.researchgate.net/publication/259484884_Harmonious_Labeling_of_Certain_Graphs">Harmonious Labeling of Certain Graphs</a>, International Journal of Advanced Engineering Research and Studies, 2 (2013), 46-48.
%F A340234 For n odd, a(n) = ceiling(n/2)^n; for n even, a(n) = ((n^2/4) + (n/2))^(n/2) (conjectured).
%e A340234 For n=3, the a(3) = 8 solutions are represented by the following adjacency matrices:
%e A340234     0  1  2         0  1  2         0  1  2         0  1  2
%e A340234 0 [ 1  1  1 ]   0 [ 1  1  0 ]   0 [ 1  0  1 ]   0 [ 1  0  0 ]
%e A340234 1 [ 1  0  0 ]   1 [ 1  1  0 ]   1 [ 0  0  0 ]   1 [ 0  1  0 ]
%e A340234 2 [ 1  0  0 ]   2 [ 0  0  0 ]   2 [ 1  0  1 ]   2 [ 0  0  1 ]
%e A340234     0  1  2         0  1  2         0  1  2         0  1  2
%e A340234 0 [ 0  1  1 ]   0 [ 0  1  0 ]   0 [ 0  0  1 ]   0 [ 0  0  0 ]
%e A340234 1 [ 1  0  1 ]   1 [ 1  1  1 ]   1 [ 0  0  1 ]   1 [ 0  1  1 ]
%e A340234 2 [ 1  1  0 ]   2 [ 0  1  0 ]   2 [ 1  1  1 ]   2 [ 0  1  1 ]
%e A340234 Notice that the number of self-loops in each graph is equal to the sum of the main diagonal.
%o A340234 (SageMath 9.2)
%o A340234 nlist = []
%o A340234 for n in range(1,162):
%o A340234     if (n % 2) == 0:
%o A340234         nlist.append(((n^2/4) + (n/2))^(n/2))
%o A340234     else:
%o A340234         nlist.append(ceil(n/2)^n)
%Y A340234 A329910 but with self-loops.
%Y A340234 For n odd, A110654 to the n-th power.
%K A340234 easy,nonn
%O A340234 1,2
%A A340234 _Patrick D. Cone_, Jan 01 2021