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A203003 Symmetric matrix based on A007598(n+1), by antidiagonals.

%I A203003 #6 Jul 12 2012 00:39:54
%S A203003 1,4,4,9,17,9,25,40,40,25,64,109,98,109,64,169,281,265,265,281,169,
%T A203003 441,740,685,723,685,740,441,1156,1933,1802,1865,1865,1802,1933,1156,
%U A203003 3025,5065,4709,4910,4819,4910,4709,5065,3025,7921,13256,12337,12827
%N A203003 Symmetric matrix based on A007598(n+1), by antidiagonals.
%C A203003 Let s=A007598(n+1) (squared Fibonacci numbers, beginning with F(2)), 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 A203003 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A203004 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%e A203003 Northwest corner:
%e A203003 1....4.....9....25....64
%e A203003 4....17....40...109...281
%e A203003 9....40....98...265...685
%e A203003 25...109...265..1865
%t A203003 s[k_] := Fibonacci[k + 1]^2;
%t A203003 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A203003 L = Transpose[U]; M = L.U; TableForm[M]
%t A203003 m[i_, j_] := M[[i]][[j]];  (* A203003 *)
%t A203003 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A203003 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]; Table[f[n], {n, 1, 12}]
%t A203003 Table[Sqrt[f[n]], {n, 1, 12}]    (* A119996 *)
%t A203003 Table[m[1, j], {j, 1, 12}]       (* A007598(n+1) *)
%Y A203003 Cf. A203004, A203001, A202453.
%K A203003 nonn,tabl
%O A203003 1,2
%A A203003 _Clark Kimberling_, Dec 27 2011