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A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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 A013915 A136445

Adjacent sequences:  A202870 A202871 A202872 * A202874 A202875 A202876

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Dec 26 2011

STATUS

approved

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Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)