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 A204134 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of f(i,j)=(2i-1 if i=j and 1 otherwise) for i>=1 and j>=1 (as in A204131). 3
 1, -1, 1, -3, 1, 3, -11, 7, -1, 21, -83, 64, -15, 1, 315, -1287, 1074, -300, 31, -1, 9765, -40527, 35067, -10570, 1287, -63, 1, 615195, -2572731, 2265129, -707539, 92653, -5313, 127, -1, 78129765, -327967227, 291222882, -92551369 (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 Table of n, a(n) for n=1..39. EXAMPLE Top of the array: 1....-1 1....-3.....1 3....-11....7....-1 21...-83....64...-15...1 MATHEMATICA f[i_, j_] := 1; f[i_, i_] := 2^(i - 1); 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}]] (* A204133 *) 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[%] (* A204134 *) TableForm[Table[c[n], {n, 1, 10}]] CROSSREFS Cf. A204133, A202605, A204016. Sequence in context: A025238 A289066 A126970 * A233168 A001351 A216021 Adjacent sequences: A204131 A204132 A204133 * A204135 A204136 A204137 KEYWORD tabl,sign AUTHOR Clark Kimberling, Jan 11 2012 STATUS approved

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Last modified June 20 18:00 EDT 2024. Contains 373532 sequences. (Running on oeis4.)