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 A204135 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the Delannoy matrix (A008288). 1
 1, -1, 2, -4, 1, 8, -28, 17, -1, 64, -384, 424, -80, 1, 1024, -10624, 19400, -7700, 401, -1, 32768, -598016, 1748224, -1225536, 161618, -2084, 1, 2097152, -68550656, 319410176, -363159040, 95891872 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 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..32. EXAMPLE Top of the array: 1....-1 2....-4.....1 8....-28....17....-1 64...-384...424...-80...1 The interlacing of zeros is illustrated by these zeros (truncated): p(1): 1 p(2): .58, 3.41 p(3): .36, 1.44, 15.19 p(4): .21, .87, 4.53, 74.3 p(5): .12, .59, 2.14, 17.22, 380.91 MATHEMATICA f[i_, 1] := 1; f[1, j_] := 1; f[i_, j_] := f[i, j - 1] + f[i - 1, j - 1] + f[i - 1, j] 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}]] (* Delannoy, A008288 *) 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, 8}] Flatten[%] (* 204135 *) TableForm[Table[c[n], {n, 1, 6}]] CROSSREFS Cf. A008288, A202605, A204016. Sequence in context: A233034 A204132 A277219 * A077387 A057551 A019823 Adjacent sequences: A204132 A204133 A204134 * A204136 A204137 A204138 KEYWORD tabl,sign AUTHOR Clark Kimberling, Jan 12 2012 STATUS approved

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Last modified June 19 16:22 EDT 2024. Contains 373503 sequences. (Running on oeis4.)