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 A204165 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of floor[(i+j)/2], as in A204164. 3
 1, -1, 1, -3, 1, -1, -2, 6, -1, 0, 4, 4, -10, 1, 0, 0, -15, -4, 15, -1, 0, 0, 0, 36, 3, -21, 1, 0, 0, 0, 0, -84, 4, 28, -1, 0, 0, 0, 0, 0, 160, -16, -36, 1, 0, 0, 0, 0, 0, 0, -300, 40, 45, -1, 0, 0, 0, 0, 0, 0, 0, 500, -75, -55, 1, 0, 0, 0, 0, 0, 0, 0, 0, -825, 130 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 and A204016 for guides to related sequences. REFERENCES (For references regarding interlacing roots, see A202605.) LINKS EXAMPLE Top of the array: 1....-1 1....-3.....1 -1....-2.....6....-1 0.....4.....4....-10...1 MATHEMATICA f[i_, j_] := Floor[(i + j)/2]; m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}] TableForm[m[8]] (* 8x8 principal submatrix *) Flatten[Table[f[i, n + 1 - i],   {n, 1, 15}, {i, 1, n}]]  (* A204164 *) p[n_] := CharacteristicPolynomial[m[n], x]; c[n_] := CoefficientList[p[n], x] TableForm[Flatten[Table[p[n], {n, 1, 10}]]] Table[c[n], {n, 1, 12}] Flatten[%]                 (* A204165 *) TableForm[Table[c[n], {n, 1, 10}]] CROSSREFS Cf. A204164, A202605, A204016. Sequence in context: A143934 A318442 A086639 * A329473 A200702 A331599 Adjacent sequences:  A204162 A204163 A204164 * A204166 A204167 A204168 KEYWORD tabl,sign AUTHOR Clark Kimberling, Jan 12 2012 STATUS approved

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Last modified January 27 15:45 EST 2022. Contains 350607 sequences. (Running on oeis4.)