OFFSET
1,2
COMMENTS
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.
...
row 1 (1,4,9,16,...) A000290
row 2 (4,17,40,73,...) A145995
diagonal (1,17,98,354,...) A000538
antidiagonal sums (1,8,35,112,...) A040977
...
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.
EXAMPLE
Northwest corner:
1.....4......9....16....25
4....17.....40....73...116
9....40.....98...184...298
16...73....184...354...584
25...116...298...584...979
MATHEMATICA
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[ Table[k^2, {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 22 2011
STATUS
approved