A139622 Triangle read by rows: T(n,k) is the number of strongly connected directed multigraphs with loops, with n arcs and k vertices.

1, 1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 10, 19, 6, 1, 1, 19, 73, 59, 9, 1, 1, 28, 208, 350, 138, 12, 1, 1, 44, 534, 1670, 1361, 301, 16, 1, 1, 60, 1215, 6476, 9724, 4364, 575, 20, 1, 1, 85, 2542, 21898, 55707, 45284, 12131, 1042, 25, 1, 1, 110, 4951, 65789, 268329, 365063, 175416, 30090, 1749, 30, 1

Offset: 1

Author

Benoit Jubin, May 01 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1
    1    1
    1    2    1
    1    6    4    1
    1   10   19    6    1
    1   19   73   59    9    1
    1   28  208  350  138   12    1
    1   44  534 1670 1361  301   16  1
    ...
T(4 edges, 2 vertices)=6: one graph 1->1, 1->1, 2->1, 1->2; one graph 1->1, 2->1, 2->1, 1->2; one graph 1->1, 1->2, 1->2, 2->1; one graph 1->1, 1->2, 2->1, 2->2; one graph 2->1, 2->1, 2->1, 1->2; one graph 1->2, 1->2, 2->1, 2->1.
T(4 edges, 3 vertices)=4: one graph 1->1, 2->1, 3->2, 1->3; one graph 2->1, 2->1, 3->2, 1->3; one graph 2->1, 3->1, 1->2, 1->3; one graph 2->1, 3->1, 1->2, 2->3.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
• R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 73.

Crossrefs

Row sums are A139627.

Cf. A136564, A139621, A143841, A350489, A350753.

Programs

PARI

\\ See PARI link in A350489 for program code.
{ my(A=A139622rows(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

Formula

T(n,1) = T(n,n) = 1.

T(n,2) = A139621(n,2) - n(n+1)/2.

Extensions

More terms from R. J. Mathar, Aug 11 2017

Terms a(34) and beyond from Andrew Howroyd, Jan 14 2022