This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A046747 #54 Feb 16 2025 08:32:39
%S A046747 1,10,338,42976,21040112,39882864736,292604283435872,
%T A046747 8286284310367538176
%N A046747 Number of n X n rational {0,1}-matrices of determinant 0.
%H A046747 J. Bourgain, V. Vu and P. M. Wood, <a href="https://doi.org/10.1016/j.jfa.2009.04.016">On the Singularity Probability of Discrete Random Matrices</a>, Journal of Functional Analysis, 258 (2010), 559-603.
%H A046747 R. P. Brent and J. H. Osborn, <a href="http://arxiv.org/abs/1208.3330">Bounds on minors of binary matrices</a>, arXiv preprint arXiv:1208.3330 [math.CO], 2012.
%H A046747 J. Kahn, J. Komlos and E. Szemeredi, <a href="https://doi.org/10.1090/S0894-0347-1995-1260107-2">On the probability that a random ±1-matrix is singular</a>, J. AMS 8 (1995), 223-240.
%H A046747 J. Komlos, <a href="http://real-j.mtak.hu/5453/">On the determinant of (0,1)-matrices</a>, Studia Math. Hungarica 2 (1967), 7-21.
%H A046747 N. Metropolis and P. R. Stein, <a href="https://doi.org/10.1016/S0021-9800(67)80006-1">On a class of (0,1) matrices with vanishing determinants</a>, J. Combin. Theory, 3 (1967), 191-198.
%H A046747 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SingularMatrix.html">Singular Matrix.</a>
%H A046747 Miodrag Zivkovic, <a href="https://arxiv.org/abs/math/0511636">Classification of small (0,1) matrices</a>, arXiv:math/0511636 [math.CO], 2005.
%H A046747 Miodrag Zivkovic, <a href="https://doi.org/10.1016/j.laa.2005.10.010">Classification of small (0,1) matrices</a>, Linear Algebra and its Applications, 414 (2006), 310-346.
%H A046747 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A046747 a(n) = 2^(n^2) - n! * binomial(2^n -1, n) + n! * A000410(n).
%F A046747 a(n) + A055165(n) = 2^(n^2) = total number of n X n (0, 1) matrices.
%F A046747 The probability that a random n X n {0,1}-matrix is singular is conjectured to be asymptotic to C(n+1, 2)*(1/2)^(n-1). [Corrected by _N. J. A. Sloane_, Jan 02 2007]
%e A046747 a(2)=10: the matrix of all 0's, 4 matrices with 2 zeros in the same row or column, 4 matrices with 3 zeros and the all-1 matrix.
%t A046747 Sum[KroneckerDelta[Det[Array[Mod[Floor[k/(2^(n*(#1-1)+#2-1))],2]&,{n,n}]],0],{k,0,(2^(n^2))-1}] (* _John M. Campbell_, Jun 24 2011 *)
%t A046747 Count[Det /@ Tuples[{0, 1}, {n, n}], 0] (* _David Trimas_, Sep 23 2024 *)
%o A046747 (PARI) A046747(n) = m=matrix(n,n); ct=0; for(x=0,2^(n*n)-1,a=binary(x+2^(n*n)); for(i=1,n, for(j=1,n,m[i,j]=a[n*i+j+1-n])); if(matdet(m)==0,ct=ct+1,); ); ct \\ _Randall L Rathbun_
%o A046747 (PARI) a(n)=sum(i=0,2^n^2-1,matdet(matrix(n,n,x,y,(i>>(n*x+y-n-1))%2))==0) \\ _Charles R Greathouse IV_, Feb 21 2015
%Y A046747 Cf. A000409, A000410, A002416, A002884, A046747, A055165, A056989, A056990.
%K A046747 hard,nonn,nice
%O A046747 1,2
%A A046747 Günter M. Ziegler (ziegler(AT)math.tu-berlin.de)
%E A046747 a(8) from _Vladeta Jovovic_, Mar 28 2006