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%I A125702 #12 Nov 03 2019 01:43:35
%S A125702 1,1,2,3,6,10,22,42,94,203,470,1082,2602,6270,15482,38525,97258,
%T A125702 247448,635910,1645411,4289010,11245670,29656148,78595028,209273780,
%U A125702 559574414,1502130920,4046853091,10939133170,29661655793
%N A125702 Number of connected categories with n objects and 2n-1 morphisms.
%C A125702 Also number of connected antitransitive relations on n objects (antitransitive meaning a R b and b R c implies not a R c); equivalently, number of free oriented bipartite trees, with all arrows going from one part to the other part.
%C A125702 Also the number of non-isomorphic multi-hypertrees of weight n - 1 with singletons allowed. A multi-hypertree with singletons allowed is a connected set multipartition (multiset of sets) with density -1, where the density of a set multipartition is the weight (sum of sizes of the parts) minus the number of parts minus the number of vertices. - _Gus Wiseman_, Oct 30 2018
%H A125702 Andrew Howroyd, <a href="/A125702/b125702.txt">Table of n, a(n) for n = 1..500</a>
%F A125702 a(n) = A122086(n) for n > 1.
%F A125702 G.f.: 2*f(x) - f(x)^2 - x where f(x) is the g.f. of A000081. - _Andrew Howroyd_, Nov 02 2019
%e A125702 From _Gus Wiseman_, Oct 30 2018: (Start)
%e A125702 Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
%e A125702   {}  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
%e A125702              {{1}{1}}  {{2}{12}}    {{13}{23}}      {{14}{234}}
%e A125702                        {{1}{1}{1}}  {{3}{123}}      {{4}{1234}}
%e A125702                                     {{1}{2}{12}}    {{2}{13}{23}}
%e A125702                                     {{2}{2}{12}}    {{2}{3}{123}}
%e A125702                                     {{1}{1}{1}{1}}  {{3}{13}{23}}
%e A125702                                                     {{3}{3}{123}}
%e A125702                                                     {{1}{2}{2}{12}}
%e A125702                                                     {{2}{2}{2}{12}}
%e A125702                                                     {{1}{1}{1}{1}{1}}
%e A125702 (End)
%o A125702 (PARI) \\ TreeGf gives gf of A000081.
%o A125702 TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o A125702 seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ _Andrew Howroyd_, Nov 02 2019
%Y A125702 Same as A122086 except for n = 1; see there for formulas. Cf. A125699.
%Y A125702 Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.
%K A125702 nonn
%O A125702 1,3
%A A125702 _Franklin T. Adams-Watters_ and _Christian G. Bower_, Jan 05 2007

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