%I #20 May 11 2020 11:22:03
%S 1,1,1,1,9,2,1,49,72,6,1,225,1250,600,24,1,961,16200,25350,5400,120,1,
%T 3969,181202,735000,470400,52920,720,1,16129,1866312,17360406,
%U 26460000,8490720,564480,5040,1,65025,18301250,362237400,1159593624,840157920,153679680,6531840,40320
%N Triangle read by rows: T(n,k) (0 <= k <= n) = k!*(Stirling2(n,k)+(k+1)*Stirling2(n,k+1))^2.
%C This is the number of Boolean matrices of dimension n and rank k having a Moore-Penrose inverse (Kim-Roush, Th. 10).
%C 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.
%H Ki Hang Kim, and Fred W. Roush, <a href="https://doi.org/10.1016/0024-3795(78)90075-7">Inverses of Boolean matrices</a>, Linear Algebra and its Applications 22 (1978): 247-262. See Th. 10.
%e Triangle begins:
%e 1,
%e 1, 1,
%e 1, 9, 2,
%e 1, 49, 72, 6,
%e 1, 225, 1250, 600, 24,
%e 1, 961, 16200, 25350, 5400, 120,
%e 1, 3969, 181202, 735000, 470400, 52920, 720,
%e 1, 16129, 1866312, 17360406, 26460000, 8490720, 564480, 5040,
%e ...
%p T := (n,k) -> k!*(Stirling2(n,k)+(k+1)*Stirling2(n,k+1))^2;
%p r:=n->[seq(T(n,k),k=0..n)];
%p for n from 0 to 12 do lprint(r(n)); od:
%Y Columns k=0-2 give: A000012, A060867, 2*A129839(n+1).
%Y Row sums give A014235.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, May 11 2020