

A202673


Array: row n shows the coefficients of the characteristic polynomial of the nth principal submatrix of the symmetric matrix A115263 based on (1,2,3,4,...); by antidiagonals.


3



1, 1, 1, 6, 1, 1, 12, 20, 1, 1, 18, 75, 50, 1, 1, 24, 166, 328, 105, 1, 1, 30, 293, 1050, 1134, 196, 1, 1, 36, 456, 2432, 5140, 3312, 336, 1, 1, 42, 655, 4690, 15471, 20814, 8514, 540, 1, 1, 48, 890, 8040, 36771, 80584
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OFFSET

1,4


COMMENTS

Let p(n)=p(n,x) be the characteristic polynomial of the nth principal submatrix of A115262 (when A115262 is formatted as a square matrix). The zeros of p(n) are positive, and they interlace the zeros of p(n+1).


LINKS

Table of n, a(n) for n=1..50.
S.G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563566.


EXAMPLE

The 1st principal submatrix (ps) of A115263 is {{1}} (using Mathematica matrix notation), with p(1)=1x and zeroset {1}.
...
The 2nd ps is {{1,2},{2,5}}, with p(2)=16x+x^2 and zeroset {0.171..., 5.828...}.
...
The 3rd ps is {{1,2,3},{2,5,8},{3,8,14}}, with p(3)=112x+20x^2x^3 and zeroset {0.099..., 0.516..., 19.383...}.
...
Top of the array:
1....1
1....6.....1
1...12....20.....1
1...18....75....50....1
1...24...166...328..105..1


MATHEMATICA

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#]  1] &[Table[k, {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]


CROSSREFS

Cf. A115262, A202671 (based on n^2), A202605 (based on Fibonacci numbers)
Sequence in context: A174345 A174449 A174150 * A202875 A203956 A082105
Adjacent sequences: A202670 A202671 A202672 * A202674 A202675 A202676


KEYWORD

tabl,sign


AUTHOR

Clark Kimberling, Dec 22 2011


STATUS

approved



