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 A002820 Number of n X n invertible binary matrices A such that A+I is invertible. (Formerly M2170 N0866) 11
 0, 2, 48, 5824, 2887680, 5821595648, 47317927329792, 1544457148312846336, 202039706313624586813440, 105823549214125066767168438272, 221819704567105547916502447159246848 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also number of linear orthomorphisms of GF(2)^n. REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..55 Zong Duo Dai, Solomon W. Golomb, and Guang Gong, Generating all linear orthomorphisms without repetition, Discrete Math. 205 (1999), 47-55. P. F. Duvall, Jr. and P. W. Harley, III, A note on counting matrices, SIAM J. Appl. Math., 20 (1971), 374-377. Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020. Kent Morrison, Matrices over F_q with no eigenvalues of 0 or 1 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. FORMULA Reference gives a recurrence. a(n) = 2^(n(n-1)/2) * A005327(n+1). MAPLE (Maple program based on Dai et al. from N. J. A. Sloane, Aug 10 2011) N:=proc(n, i) option remember; if i = 1 then 1 else (2^n-2^(i-1))*N(n, i-1); fi; end; Oh:=proc(n) option remember; local r; global N; if n=0 then RETURN(1) elif n=1 then RETURN(0) else add( 2^(r-2)*N(n, r)*2^(r*(n-r))*Oh(n-r), r=2..n); fi; end; [seq(Oh(n), n=1..15)]; MATHEMATICA 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 *) CROSSREFS Cf. A002884. Sequence in context: A067626 A053071 A238838 * A196448 A053290 A056989 Adjacent sequences: A002817 A002818 A002819 * A002821 A002822 A002823 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Mar 17 2000 Entry revised by N. J. A. Sloane, Aug 10 2011 STATUS approved

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Last modified December 6 14:43 EST 2022. Contains 358644 sequences. (Running on oeis4.)