A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals.

1, 3, 3, 7, 10, 7, 15, 24, 24, 15, 31, 52, 59, 52, 31, 63, 108, 129, 129, 108, 63, 127, 220, 269, 284, 269, 220, 127, 255, 444, 549, 594, 594, 549, 444, 255, 511, 892, 1109, 1214, 1245, 1214, 1109, 892, 511, 1023, 1788, 2229, 2454, 2547, 2547, 2454

Offset: 1

Author

Clark Kimberling, Dec 26 2011

Keywords

• nonn
• tabl

Comments

Let s=(1,3,7,15,31,...) 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 A202873 represents the matrix product M=T'*T. M is the self-fusion matrix of s, as defined at A193722. See A202767 for characteristic polynomials of principal submatrices of M.

Examples

			Northwest corner:
1.....3.....7...15...31.....63
3....10....24...52...108...220
7....24....59..129...269...549
15...52...129..284...594..1214
31...108..269..594..1245..2547
		

Crossrefs

Cf. A202767.

Programs

Mathematica

s[k_] := -1 + 2^k;
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
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}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}] (* A000295, Eulerian *)
Table[m[1, j], {j, 1, 12}]    (* A000225 *)
Table[m[2, j], {j, 1, 12}]    (* A053208 *)