A202670 - OEIS

%I #7 Jul 12 2012 00:39:53

%S 1,4,4,9,17,9,16,40,40,16,25,73,98,73,25,36,116,184,184,116,36,49,169,

%T 298,354,298,169,49,64,232,440,584,584,440,232,64,81,305,610,874,979,

%U 874,610,305,81,100,388,808,1224,1484,1484,1224,808,388,100,121

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

%C 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 ...

%C row 1 (1,4,9,16,...) A000290

%C row 2 (4,17,40,73,...) A145995

%C diagonal (1,17,98,354,...) A000538

%C antidiagonal sums (1,8,35,112,...) A040977

%C ...

%C 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 Northwest corner:

%e 1.....4......9....16....25

%e 4....17.....40....73...116

%e 9....40.....98...184...298

%e 16...73....184...354...584

%e 25...116...298...584...979

%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];

%t L = Transpose[U]; M = L.U; TableForm[M]

%t m[i_, j_] := M[[i]][[j]];

%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

%Y Cf. A000290, A202671, A193722.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 22 2011