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

%I A203951 #6 Jul 12 2012 00:39:54
%S A203951 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,
%T A203951 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,
%U A203951 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
%N A203951 Symmetric matrix based on (1,0,0,0,1,0,0,0,...), by antidiagonals.
%C A203951 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.
%e A203951 Northwest corner:
%e A203951 1 0 0 0 1 0 0 0 1 0
%e A203951 0 1 0 0 0 1 0 0 0 1
%e A203951 0 0 1 0 0 0 1 0 0 0
%e A203951 0 0 0 1 0 0 0 1 0 0
%e A203951 1 0 0 0 2 0 0 0 2 0
%e A203951 0 1 0 0 0 2 0 0 0 2
%e A203951 0 0 1 0 0 0 2 0 0 0
%e A203951 0 0 0 1 0 0 0 2 0 0
%e A203951 1 0 0 0 2 0 0 0 3 0
%t A203951 t = {1, 0, 0, 0}; t1 = Flatten[{t, t, t, t, t, t, t, t}];
%t A203951 f[k_] := t1[[k]];
%t A203951 U[n_] :=
%t A203951   NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[
%t A203951    Table[f[k], {k, 1, n}]];
%t A203951 L[n_] := Transpose[U[n]];
%t A203951 p[n_] := CharacteristicPolynomial[L[n].U[n], x];
%t A203951 c[n_] := CoefficientList[p[n], x]
%t A203951 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A203951 Table[c[n], {n, 1, 12}]
%t A203951 Flatten[%]  (* A203952 *)
%Y A203951 Cf. A203951, A202453.
%K A203951 nonn,tabl
%O A203951 1,41
%A A203951 _Clark Kimberling_, Jan 08 2012