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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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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.
LINKS
Ki Hang Kim, and Fred W. Roush, Inverses of Boolean matrices, Linear Algebra and its Applications 22 (1978): 247-262. See Th. 10.
EXAMPLE
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,
...
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:
CROSSREFS
Columns k=0-2 give: A000012, A060867, 2*A129839(n+1).
Row sums give A014235.
Sequence in context: A010536 A239908 A293171 * A335086 A151898 A080994
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, May 11 2020
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)