A125702 Number of connected categories with n objects and 2n-1 morphisms.

1, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793

Offset: 1

Author

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Keywords

• nonn

Comments

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.

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

Examples

			From _Gus Wiseman_, Oct 30 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(6) = 10 multi-hypertrees of weight n - 1 with singletons allowed:
  {}  {{1}}  {{12}}    {{123}}      {{1234}}        {{12345}}
             {{1}{1}}  {{2}{12}}    {{13}{23}}      {{14}{234}}
                       {{1}{1}{1}}  {{3}{123}}      {{4}{1234}}
                                    {{1}{2}{12}}    {{2}{13}{23}}
                                    {{2}{2}{12}}    {{2}{3}{123}}
                                    {{1}{1}{1}{1}}  {{3}{13}{23}}
                                                    {{3}{3}{123}}
                                                    {{1}{2}{2}{12}}
                                                    {{2}{2}{2}{12}}
                                                    {{1}{1}{1}{1}{1}}
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

Same as A122086 except for n = 1; see there for formulas. Cf. A125699.

Cf. A000081, A000272, A007716, A007717, A030019, A052888, A134954, A317631, A317632, A318697, A320921, A321155.

Programs

PARI

\\ TreeGf gives gf of A000081.
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)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2 - x)} \\ Andrew Howroyd, Nov 02 2019

Formula

a(n) = A122086(n) for n > 1.

G.f.: 2*f(x) - f(x)^2 - x where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019