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%I #26 Oct 18 2023 09:21:51
%S 1,2,20,640,97824,47545088
%N Number of n X n (-1, 1)-matrices which have only eigenvalues with strictly negative real part (which implies that the matrix has all nonzero eigenvalues).
%C As this problem is symmetric with sign we can get the same numbers for strictly positive real parts.
%C All values for n > 1 are even, because a transposed matrix has the same spectrum of eigenvalues.
%C Matrices with determinant 0 are not counted.
%C Let M be such a matrix then the limit of ||exp(t*M)*y|| if t goes to infinity will be zero.
%C n = 5 is the first case where not all entries on the main diagonal are -1. 93984 matrices with 5 times -1 on the main diagonal and 5*768 with 4 times -1 on the main diagonal have only eigenvalues with strictly negative real part.
%C In the case n = 6, 43586048 matrices with 6 times -1 on the main diagonal, 6*656000 matrices with 5 times -1 on the main diagonal and 15*1536 matrices with 5 times -1 on the main diagonal have only eigenvalues with strictly negative real part.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%e For n = 2 the matrices are:
%e .
%e -1, 1
%e -1, -1
%e .
%e -1, -1
%e 1, -1.
%Y Cf. A056990.
%Y Cf. A083058, A085506, A087488, A098148, A086510, A207259.
%Y Cf. A219736, A346210, A271570, A271588, A296605, A306002.
%Y Cf. A306791, A306792, A306793, A306794, A306795, A326928.
%Y Cf. A346209.
%K nonn,more
%O 1,2
%A _Thomas Scheuerle_, Aug 12 2023