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A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.

%I A202670 #7 Jul 12 2012 00:39:53
%S A202670 1,4,4,9,17,9,16,40,40,16,25,73,98,73,25,36,116,184,184,116,36,49,169,
%T A202670 298,354,298,169,49,64,232,440,584,584,440,232,64,81,305,610,874,979,
%U A202670 874,610,305,81,100,388,808,1224,1484,1484,1224,808,388,100,121
%N A202670 Symmetric matrix based on A000290 (the squares), by antidiagonals.
%C A202670 Let s=(1,4,9,16,...) 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 A202670 represents the matrix product M=T'*T.  M is the self-fusion matrix of s, as defined at A193722.  See A202671 for characteristic polynomials of principal submatrices of M.
%C A202670 ...
%C A202670 row 1 (1,4,9,16,...) A000290
%C A202670 row 2 (4,17,40,73,...) A145995
%C A202670 diagonal (1,17,98,354,...) A000538
%C A202670 antidiagonal sums (1,8,35,112,...) A040977
%C A202670 ...
%C A202670 The n-th "square border sum" m(n,1)+m(n,2)+...+m(n,n)+m(n-1,n)+m(n-2,n)+...+m(1,n) is a squared square pyramidal number: [n*(n+1)*(2*n+1)/6]^2; see A000330.
%e A202670 Northwest corner:
%e A202670 1.....4......9....16....25
%e A202670 4....17.....40....73...116
%e A202670 9....40.....98...184...298
%e A202670 16...73....184...354...584
%e A202670 25...116...298...584...979
%t A202670 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];
%t A202670 L = Transpose[U]; M = L.U; TableForm[M]
%t A202670 m[i_, j_] := M[[i]][[j]];
%t A202670 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%Y A202670 Cf. A000290, A202671, A193722.
%K A202670 nonn,tabl
%O A202670 1,2
%A A202670 _Clark Kimberling_, Dec 22 2011