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A002820 Number of n X n invertible binary matrices A such that A+I is invertible.

%I A002820 M2170 N0866 #56 Jan 10 2025 18:14:52
%S A002820 1,0,2,48,5824,2887680,5821595648,47317927329792,1544457148312846336,
%T A002820 202039706313624586813440,105823549214125066767168438272,
%U A002820 221819704567105547916502447159246848,1860304261534304703934696550224148083769344,62413833036707798343389591015829588620560344023040
%N A002820 Number of n X n invertible binary matrices A such that A+I is invertible.
%C A002820 Also number of linear orthomorphisms of GF(2)^n.
%D A002820 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002820 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002820 Vincenzo Librandi, <a href="/A002820/b002820.txt">Table of n, a(n) for n = 0..55</a>
%H A002820 Zong Duo Dai, Solomon W. Golomb, and Guang Gong, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00035-7">Generating all linear orthomorphisms without repetition</a>, Discrete Math. 205 (1999), 47-55.
%H A002820 P. F. Duvall, Jr. and P. W. Harley, III, <a href="http://www.jstor.org/stable/2099958">A note on counting matrices</a>, SIAM J. Appl. Math., 20 (1971), 374-377.
%H A002820 Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.
%H A002820 Kent Morrison, <a href="http://www.calpoly.edu/~kmorriso/Research/mnev01.pdf">Matrices over F_q with no eigenvalues of 0 or 1</a>
%H A002820 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A002820 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A002820 Reference gives a recurrence.
%F A002820 a(n) = 2^(n(n-1)/2) * A005327(n+1).
%p A002820 # (Maple program based on Dai et al. from _N. J. A. Sloane_, Aug 10 2011)
%p A002820 N:=proc(n,i) option remember; if i = 1 then 1 else (2^n-2^(i-1))*N(n,i-1); fi; end;
%p A002820 Oh:=proc(n) option remember; local r; global N;
%p A002820 if n=0 then RETURN(1) elif n=1 then RETURN(0) else
%p A002820 add( 2^(r-2)*N(n,r)*2^(r*(n-r))*Oh(n-r), r=2..n); fi; end;
%p A002820 [seq(Oh(n),n=0..15)];
%t A002820 ni[n_, i_] := ni[n, i] = If[i == 1, 1, (2^n - 2^(i-1))*ni[n, i-1]]; a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[ 2^(r-2)*ni[n, r]*2^(r*(n-r))*a[n-r], {r, 2, n}]; Table[a[n], {n, 1, 11}] (* _Jean-François Alcover_, Jan 19 2012, after Maple *)
%Y A002820 Cf. A002884.
%Y A002820 Column k=0 of A346201 and of A346381.
%K A002820 nonn,nice,easy
%O A002820 0,3
%A A002820 _N. J. A. Sloane_
%E A002820 More terms from _Vladeta Jovovic_, Mar 17 2000
%E A002820 Entry revised by _N. J. A. Sloane_, Aug 10 2011
%E A002820 a(0)=1 prepended by _Alois P. Heinz_, Jan 10 2025