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A203906 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of A203905. 3
1, -1, 1, -2, 1, 1, -4, 4, -1, 1, -6, 11, -6, 1, 1, -8, 22, -24, 9, -1, 1, -10, 37, -62, 46, -12, 1, 1, -12, 56, -128, 148, -80, 16, -1, 1, -14, 79, -230, 367, -314, 130, -20, 1, 1, -16, 106, -376, 771, -920, 610, -200, 25, -1, 1, -18, 137 (list; 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 positive, and they interlace the zeros of p(n+1).  See A202605 for a guide to related sequences.

If we omit the main diagonal of this array and ignore the signs of the entries then the resulting array, reading the rows in reverse order, appears to equal the Riordan array (1/((1 + x)*(1 - x)^3), x/(1 - x)^2), whose generating function begins 1 + (2 + t)*x + (4 + 4*t + t^2)*x^2 + (6 + 11*t + 6*t^2 + t^3)*x^3 + (9 + 24*t + 22*t^2 + 8*t^3 + t^4)*x^4 + .... - Peter Bala, Sep 17 2019

REFERENCES

(For references regarding interlacing roots, see A202605.)

LINKS

Table of n, a(n) for n=1..57.

EXAMPLE

Top of the array:

1...-1

1...-2....1

1...-4....4...-1

1...-6...11...-6....1

1...-8...22...-24...9...-1

MATHEMATICA

t = {1, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t, t, t}];

f[k_] := t1[[k]];

U[n_] := NestList[Most[Prepend[#, 0]] &, #,

Length[#] - 1] &[Table[f[k], {k, 1, n}]];

L[n_] := Transpose[U[n]];

p[n_] := CharacteristicPolynomial[L[n].U[n], x];

c[n_] := CoefficientList[p[n], x]

TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

Table[c[n], {n, 1, 12}]

Flatten[%]                         (* A203906 *)

TableForm[Table[c[n], {n, 1, 10}]]

Table[p[n] /. x -> -1, {n, 1, 16}] (* A166516 *)

CROSSREFS

Cf. A202605, A166516.

Sequence in context: A259698 A274643 A172991 * A274310 A096806 A116672

Adjacent sequences:  A203903 A203904 A203905 * A203907 A203908 A203909

KEYWORD

tabf,sign

AUTHOR

Clark Kimberling, Jan 08 2012

STATUS

approved

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Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)