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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
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
Sequence in context: A233034 A204132 A277219 * A077387 A057551 A019823
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.)