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Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).
4

%I #9 Mar 17 2019 12:26:28

%S 1,1,1,1,1,1,2,1,6,1,10,1,19,1,28,1,1,44,2,1,60,10,1,85,31,1,110,90,1,

%T 146,222,1,182,520,1,231,1090,1,1,280,2180,2,1,344,4090,11,1

%N Table read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops and no vertex of degree 0, with n arcs and k vertices, which are transitive (the existence of a path between two points implies the existence of an arc between those two points).

%C Length of the n^th row: floor(sqrt(n)).

%C These graphs are reflexive (each vertex has a self-loop), so T(n,k) = sum(A139621(n-k^2,m),m=0..k)

%C T(n,1) = 1, T(n,2) = A005993(n-4), T(n,3) = A050927(n-9), T(n,4) = A050929(n-16).

%C Row sums: A139630.

%e Triangle begins:

%e 1

%e 1

%e 1

%e 1 1

%e 1 2

%e 1 6

%e 1 10

%e 1 19

%e 1 28 1

%Y Cf. A139623, A139624.

%K nonn,tabf

%O 1,7

%A _Benoit Jubin_, May 01 2008, Sep 01 2008