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A064231 Triangle read by rows: T(n,k) = number of rational (+1,-1) matrices of rank k (n >= 1, 1 <= k <= n). 4

%I

%S 2,8,8,32,288,192,128,6272,36864,22272,512,115200,3456000,18432000,

%T 11550720,2048,1968128,243302400,6471168000,36373708800,25629327360,

%U 8192,32514048,14809546752,1557061632000,43378316083200,281770208133120

%N Triangle read by rows: T(n,k) = number of rational (+1,-1) matrices of rank k (n >= 1, 1 <= k <= n).

%C Rows add to 2^(n^2) = A002416. - _Jonathan Vos Post_, Feb 27 2011

%C Komlos and later Kahn, Komlos and Szemeredi show that almost all such matrices are invertible.

%D J. Kahn, J. Komlos and E. Szemeredi: On the probability that a random +-1 matrix is singular, J. AMS 8 (1995), 223-240.

%D J. Komlos, On the determinants of random matrices, Studia Sci. Math. Hungar., 3 (1968), 387-399.

%e 2; 8,8; 32,288,192; 128,6272,36864,22272; ...

%o (PARI) T=matrix(4,4); { for(n=1,4, mm=matrix(n,n); for(k=1,n, T[n,k]=0); forvec(x=vector(n*n,i,[0,1]), for(i=1,n, for(j=1,n,mm[i,j]=(-1)^x[i+n*(j-1)])); T[n,matrank(mm)]++); for(k=1,n,print1(T[n,k], if(k<n,",","; ")));) }

%Y Cf. A064230, A000409, A000410, A046747.

%K nonn,nice,tabl

%O 1,1

%A _N. J. A. Sloane_, Sep 23 2001

%E More terms and PARI code from _Michael Somos_, Sep 25 2001

%E More terms from _Vladeta Jovovic_, Apr 02 2006

%E Offset changed to 1 by _T. D. Noe_, Mar 02 2011

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Last modified November 15 04:00 EST 2018. Contains 317225 sequences. (Running on oeis4.)