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A204181
Symmetric matrix based on f(i,j) defined by f(i,1)=f(1,j)=1; f(i,i)= 2i-1; f(i,j)=0 otherwise; by antidiagonals.
3
1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 1, 0, 5, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 7, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 9, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0
OFFSET
1,5
COMMENTS
A204181 represents the matrix M given by f(i,j) for i>=1 and j>=1. See A204182 for characteristic polynomials of principal submatrices of M, with interlacing zeros. See A204016 for a guide to other choices of M.
EXAMPLE
Northwest corner:
1 1 1 1 1 1 1 1
1 3 0 0 0 0 0 0
1 0 5 0 0 0 0 0
1 0 0 7 0 0 0 0
1 0 0 0 9 0 0 0
MATHEMATICA
f[i_, j_] := 0; f[1, j_] := 1;
f[i_, 1] := 1; f[i_, i_] := 2 i - 1;
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}]] (* A204181 *)
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, 12}]
Flatten[%] (* A204182 *)
TableForm[Table[c[n], {n, 1, 10}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 12 2012
STATUS
approved