A085340 a(n) is the value of determinant of the following special matrix: diagonal values equal to n-2; upper triangular entries equal to -1; lower triangular values are +1.

-1, 1, 4, 41, 528, 8177, 148160, 3077713, 72147712, 1884629825, 54294967296, 1710428956601, 58496602689536, 2158563109641265, 85487558566199296, 3616912482448035233, 162819625954342010880, 7770488166051562690817, 391896604540625999888384

Offset: 1

Author

Labos Elemer, Jul 08 2003

Keywords

• sign

Comments

These invertible matrices are used in formal neural network theory to generate transient-free state transition graphs with using suitable threshold vectors.

Examples

			n=5: matrix =
+3,-1,-1,-1,-1
+1,+3,-1,-1,-1
+1,+1,+3,-1,-1
+1,+1,+1,+3,-1
+1,+1,+1,+1,+3,
with determinant=528=a(5). a(1)=-1 is the only negative term.
		

References

• Labos E.: The most complicated networks of formal neurons. In Proc. of IEEE. first International Conference on Neural Networks. San Diego,USA,1987.[Editors: Caudill,M. and Butler Ch.]; Vol. III, pp. 301-308.

Programs

Mathematica

f[x_, y_] := Sign[y-x] g[x_, y_, z_] := (z-2)*(1-Abs[f[x, y]]); a=Table[Table[f[w, s], {w, 1, q}], {s, 1, q}]; b=Table[Table[g[w, s, q], {w, 1, q}], {s, 1, q}]; m=matrix=a+b; Det[m]; Table[Det[Table[Table[f[w, s]+g[w, s, q], {w, 1, q}], {s, 1, q}]], {q, 1, 20}]