A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,41

Comments

Let s be the periodic sequence (1,0,0,0,1,0,0,0,...) and let T be the infinite square matrix whose n-th row is formed by putting n-1 zeros before the terms of s. Let T' be the transpose of T. Then A203951 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A203952 for characteristic polynomials of principal submatrices of M, with interlacing zeros.

Links

Table of n, a(n) for n=1..99.

Example

Northwest corner:
1 0 0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0 0 1
0 0 1 0 0 0 1 0 0 0
0 0 0 1 0 0 0 1 0 0
1 0 0 0 2 0 0 0 2 0
0 1 0 0 0 2 0 0 0 2
0 0 1 0 0 0 2 0 0 0
0 0 0 1 0 0 0 2 0 0
1 0 0 0 2 0 0 0 3 0

Mathematica

t = {1, 0, 0, 0}; t1 = Flatten[{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[%]  (* A203952 *)

Crossrefs

Cf. A203951, A202453.

Sequence in context: A322549 A349645 A237996 * A323591 A105348 A016406

Adjacent sequences: A203948 A203949 A203950 * A203952 A203953 A203954

Keyword

nonn,tabl

Author

Clark Kimberling, Jan 08 2012

Status

approved