A143841 Table read by antidiagonals: T(n,k) is the number of strongly connected directed multigraphs with loops with n arcs and up to k vertices.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 7, 1, 0, 1, 1, 2, 4, 11, 11, 1, 0, 1, 1, 2, 4, 12, 30, 20, 1, 0, 1, 1, 2, 4, 12, 36, 93, 29, 1, 0, 1, 1, 2, 4, 12, 37, 152, 237, 45, 1, 0, 1, 1, 2, 4, 12, 37, 161, 587, 579, 61, 1, 0

Offset: 0

Author

Benoit Jubin, Sep 02 2008

Keywords

• nonn
• tabl

Examples

			Array begins:
=============================================
n\k | 0 1  2   3    4    5    6    7    8
----+----------------------------------------
  0 | 1 1  1   1    1    1    1    1    1 ...
  1 | 0 1  1   1    1    1    1    1    1 ...
  2 | 0 1  2   2    2    2    2    2    2 ...
  3 | 0 1  3   4    4    4    4    4    4 ...
  4 | 0 1  7  11   12   12   12   12   12 ...
  5 | 0 1 11  30   36   37   37   37   37 ...
  6 | 0 1 20  93  152  161  162  162  162 ...
  7 | 0 1 29 237  587  725  737  738  738 ...
  8 | 0 1 45 579 2249 3610 3911 3927 3928 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..860

Crossrefs

Partial sums of the rows of A139622.

Main diagonal is A139627.

Cf. A138107, A129620, A143842, A350489.

Programs

PARI

\\ See PARI link in A350489 for program code.
A(n)={my(data=A139622rows(n), M=matrix(n+1, n+1, i, j, if(i==1, 1, sum(k=1, min(i-1,j-1), data[i-1][k])))); M}
{ my(M=A(8)); for(n=1, #M~, print(M[n,])) } \\ Andrew Howroyd, Jan 14 2022

Formula

T(n,k) = Sum_{p=0..k} A139622(n,p).

T(n,k) = A139627(n) for k >= n.

T(n,2) = A129620(n,2) - n*(n-1)/2.

Extensions

Name clarified and terms a(32) and beyond from Andrew Howroyd, Jan 14 2022