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A334613 Number of distinct graceful labelings of bipartite graphs with n vertices and n-1 edges.

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%I A334613 #22 Oct 19 2020 13:04:11
%S A334613 1,1,1,2,7,28,151,907,6467,51151,458478,4455476,48162974,557022350,
%T A334613 7020022164,94129389215,1357181821028,20661645476348
%N A334613 Number of distinct graceful labelings of bipartite graphs with n vertices and n-1 edges.
%C A334613 Consider vertices numbered 1 to n. Add the edges 1--n, 2--n, and either 1--(n+1-k), 2--(n+2-k), ... or k--n for 3<=k<n. (Altogether (n-1)!/2 possibilities.) If the resulting graph has no odd cycles, this is a graceful labeling of a bipartite graph.
%e A334613 For example, the seven labelings for n=5 are:
%e A334613   1--5, 2--5, 1--3, 2--3;
%e A334613   1--5, 2--5, 1--3, 3--4;
%e A334613   1--5, 2--5, 1--3, 4--5;
%e A334613   1--5, 2--5, 2--4, 2--3;
%e A334613   1--5, 2--5, 2--4, 3--4;
%e A334613   1--5, 2--5, 3--5, 3--4;
%e A334613   1--5, 2--5, 3--5, 4--5.
%e A334613 The first of these is the four-cycle plus an isolated vertex; the other six are the trees in A337274.
%Y A334613 Cf. A337274.
%K A334613 nonn,more
%O A334613 1,4
%A A334613 _Don Knuth_, Sep 08 2020
%E A334613 a(17)-a(18) from _Bert Dobbelaere_, Sep 11 2020