A136564 Array read by rows: T(n,k) is the number of directed multigraphs with loops with n arcs, k vertices, and no vertex of degree 0.

1, 1, 1, 5, 4, 1, 1, 9, 21, 16, 4, 1, 1, 18, 71, 108, 71, 22, 4, 1, 1, 27, 194, 491, 557, 326, 101, 22, 4, 1, 1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1, 1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1, 1, 84, 2095, 18823, 72064

Offset: 1

Author

Benoit Jubin, Apr 14 2008

Keywords

• nonn
• tabf

Comments

Length of the n^th row: 2n.

Examples

			1, 1;
1, 5, 4, 1;
1, 9, 21, 16, 4, 1;
1, 18, 71, 108, 71, 22, 4, 1;
1, 27, 194, 491, 557, 326, 101, 22, 4, 1;
1, 43, 476, 1903, 3353, 3062, 1587, 497, 111, 22, 4, 1;
1, 59, 1030, 6298, 16644, 22352, 17035, 7982, 2433, 555, 111, 22, 4, 1;
		

Crossrefs

Row sums: A052171. Partial row sums: A138107.

Sums of the first m entries of each row: A005993 (m=2), A050927 (m=3), A050929 (m=4).

Formula

T(n,1) = 1 if n > 0.

T(n,2n) = 1 if n > 0.

T(n,2n-1) = 4 if n >= 2.

T(n,2n-k) = A144047(k) for n large enough (conjecturally, n >= 2k is enough).

T(n,2) = (n^3 + 6*n^2 + 11*n - 6)/12 + ((n+2)/4)[n even]. (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012

Extensions

More terms from Benoit Jubin and Vladeta Jovovic, Sep 08 2008