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.)
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
KEYWORD
tabf,sign
AUTHOR
Clark Kimberling, Jan 08 2012
STATUS
approved