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A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.

%I A327904 #11 Oct 03 2019 16:54:48
%S A327904 1,1,2,8,48,432,5184,82944,1658880,41472000,1244160000,44789760000,
%T A327904 1881169920000,92177326080000,5161930260480000,330363536670720000,
%U A327904 23786174640291840000,1926680145863639040000,173401213127727513600000,17340121312772751360000000
%N A327904 Number of labeled simple graphs with vertices {1..n} such that every edge has a different sum.
%H A327904 Andrew Howroyd, <a href="/A327904/b327904.txt">Table of n, a(n) for n = 0..100</a>
%H A327904 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%H A327904 Gus Wiseman, <a href="/A327904/a327904.png">The a(4) = 48 graphs where every edge has a different sum.</a>
%F A327904 a(n) = Product_{k=1..2*n+1} ceiling(k/4). - _Andrew Howroyd_, Oct 02 2019
%e A327904 The graph with edge-set {{1,2},{1,3},{1,4},{2,3}}, which looks like a triangle with a tail, has edges {1,4} and {2,3} with equal sum, so is not counted under a(4).
%p A327904 a:= proc(n) option remember; `if`(n=0, 1,
%p A327904       a(n-1)*ceil(n/2)*ceil(n/2+1/4))
%p A327904     end:
%p A327904 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 03 2019
%t A327904 stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t A327904 qes[n_]:=stableSets[Subsets[Range[n],{2}],Total[#1]==Total[#2]&];
%t A327904 Table[Length[qes[n]],{n,0,5}]
%o A327904 (PARI) a(n) = {prod(k=1, 2*n+1, ceil(k/4))} \\ _Andrew Howroyd_, Oct 02 2019
%Y A327904 The generalization to antichains is A326030.
%Y A327904 Cf. A006125, A006129, A275780, A321469, A326519, A326535, A327903.
%K A327904 nonn
%O A327904 0,3
%A A327904 _Gus Wiseman_, Sep 30 2019
%E A327904 Terms a(8) and beyond from _Andrew Howroyd_, Oct 02 2019