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

1,2

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.

Links

Table of n, a(n) for n=1..52.

Example

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

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 *)

Crossrefs

Cf. A202767.

Sequence in context: A117525 A075149 A161618 * A157933 A350394 A013915

Adjacent sequences: A202870 A202871 A202872 * A202874 A202875 A202876

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 26 2011

Status

approved