

A125702


Number of connected categories with n objects and 2n1 morphisms.


6



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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 nonisomorphic multihypertrees of weight n  1 with singletons allowed. A multihypertree 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


LINKS

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


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


EXAMPLE

From Gus Wiseman, Oct 30 2018: (Start)
Nonisomorphic representatives of the a(1) = 1 through a(6) = 10 multihypertrees 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)


PROG

(PARI) \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[nk+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n)  TreeGf(n)^2  x)} \\ Andrew Howroyd, Nov 02 2019


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.
Sequence in context: A049527 A074371 A032202 * A052817 A156803 A002992
Adjacent sequences: A125699 A125700 A125701 * A125703 A125704 A125705


KEYWORD

nonn


AUTHOR

Franklin T. AdamsWatters and Christian G. Bower, Jan 05 2007


STATUS

approved



