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A293147 Triangle read by rows: coefficients of the characteristic polynomial of the n-th submatrix of A191898.

%I A293147 #33 Oct 22 2017 11:50:27
%S A293147 0,1,-1,-2,0,1,6,4,-2,-1,0,-12,-5,3,1,0,60,49,-3,-7,-1,0,360,84,-90,
%T A293147 -19,5,1,0,-2520,-1308,414,241,-5,-11,-1,0,0,3780,1752,-590,-290,9,12,
%U A293147 1,0,0,0,-7560,-2874,1122,406,-19,-14,-1
%N A293147 Triangle read by rows: coefficients of the characteristic polynomial of the n-th submatrix of A191898.
%C A293147 It appears that for n > 10, the nearest integer to the largest negative eigenvalue of the n-th characteristic polynomial is equal to the previous prime sequence A007917(n).
%C A293147 A007917(n) = round(max(-eigenvalues(A191898(1..n,1..n)))) (for n > 10), has been verified in the range n=11 to n=100.
%H A293147 Mats Granvik, <a href="https://math.stackexchange.com/questions/156035/primes-approximated-by-eigenvalues">Primes approximated by eigenvalues?</a>
%H A293147 Mats Granvik, <a href="/A293147/a293147.txt">Mathematica program to verify the agreement between the largest negative eigenvalues and the previous prime sequence.</a>
%e A293147    0;
%e A293147    1,    -1;
%e A293147   -2,     0,     1;
%e A293147    6,     4,    -2,    -1;
%e A293147    0,   -12,    -5,     3,     1;
%e A293147    0,    60,    49,    -3,    -7,   -1;
%e A293147    0,   360,    84,   -90,   -19,    5,   1;
%e A293147    0, -2520, -1308,   414,   241,   -5, -11,  -1;
%e A293147    0,     0,  3780,  1752,  -590, -290,   9,  12,   1;
%e A293147    0,     0,     0, -7560, -2874, 1122, 406, -19, -14, -1;
%e A293147    ...
%e A293147 max(-eigenvalues(A191898(1..12,1..12)))=11.096...
%e A293147 max(-eigenvalues(A191898(1..13,1..13)))=12.9021...
%t A293147 Clear[A,B,nnn]; nnn=9; charpol = Table[A = Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, nn}], {n, 1, nn}]; B = Table[Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {k, 1, nn}], {n, 1, nn}]; CoefficientList[CharacteristicPolynomial[A.B, x], x], {nn, 1, nnn}];Flatten[charpol]
%Y A293147 Cf. A191898, A007917.
%K A293147 sign,tabl
%O A293147 0,4
%A A293147 _Mats Granvik_, Oct 01 2017