A334689 Triangle read by rows: T(n,k) (0 <= k <= n) = k!*(Stirling2(n,k)+(k+1)*Stirling2(n,k+1))^2.

1, 1, 1, 1, 9, 2, 1, 49, 72, 6, 1, 225, 1250, 600, 24, 1, 961, 16200, 25350, 5400, 120, 1, 3969, 181202, 735000, 470400, 52920, 720, 1, 16129, 1866312, 17360406, 26460000, 8490720, 564480, 5040, 1, 65025, 18301250, 362237400, 1159593624, 840157920, 153679680, 6531840, 40320

Offset: 0

Author

N. J. A. Sloane, May 11 2020

Keywords

• nonn
• tabl

Comments

This is the number of Boolean matrices of dimension n and rank k having a Moore-Penrose inverse (Kim-Roush, Th. 10).

Theorem 8 of the same Kim-Roush paper gives a formula for the number of Boolean matrices of dimension n and rank k having a minimum-norm g-inverse. Unfortunately the formula appears to produce negative numbers.

Examples

			Triangle begins:
1,
1, 1,
1, 9, 2,
1, 49, 72, 6,
1, 225, 1250, 600, 24,
1, 961, 16200, 25350, 5400, 120,
1, 3969, 181202, 735000, 470400, 52920, 720,
1, 16129, 1866312, 17360406, 26460000, 8490720, 564480, 5040,
...
		

Links

• Ki Hang Kim, and Fred W. Roush, Inverses of Boolean matrices, Linear Algebra and its Applications 22 (1978): 247-262. See Th. 10.

Crossrefs

Columns k=0-2 give: A000012, A060867, 2*A129839(n+1).

Row sums give A014235.

Programs

Maple

T := (n,k) -> k!*(Stirling2(n,k)+(k+1)*Stirling2(n,k+1))^2;
r:=n->[seq(T(n,k),k=0..n)];
for n from 0 to 12 do lprint(r(n)); od: