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%I #15 Dec 31 2024 14:37:22
%S 1,1,2,14,174,3655,122439,6470268
%N Total numbers of fundamentally distinct graceful labelings of all simple graphs on n vertices.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>.
%e In the below, G: n stands for "G has n fundamentally distinct graceful labelings".
%e a(1) = 1 since K_1: 1.
%e a(2) = 1 since P_2: 1.
%e a(3) = 2 since P_3: 1, C_3: 1.
%e a(4) = 14 since C_3+K_1: 1, K_1,3 (claw): 1, diamond: 4, P_4: 1, paw: 5, C_4: 1, K_4: 1
%t {1, 1} ~ Join ~ Table[Total[GraphData[#, "GracefulLabelingCount"] & /@ GraphData["Graceful", n]], {n, 3, 7}]
%Y Cf. A333727 (totals of all graceful labelings of simple graphs on n vertices).
%Y Cf. A379575 (totals of all fundamentally distinct graceful labelings of simple graphs on n nodes containing no isolated points).
%K nonn,more,new
%O 1,3
%A _Eric W. Weisstein_, Dec 26 2024