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

%I A202873 #9 Jul 12 2012 00:39:53
%S A202873 1,3,3,7,10,7,15,24,24,15,31,52,59,52,31,63,108,129,129,108,63,127,
%T A202873 220,269,284,269,220,127,255,444,549,594,594,549,444,255,511,892,1109,
%U A202873 1214,1245,1214,1109,892,511,1023,1788,2229,2454,2547,2547,2454
%N A202873 Symmetric matrix based on (1,3,7,15,31,...), by antidiagonals.
%C A202873 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.
%e A202873 Northwest corner:
%e A202873 1.....3.....7...15...31.....63
%e A202873 3....10....24...52...108...220
%e A202873 7....24....59..129...269...549
%e A202873 15...52...129..284...594..1214
%e A202873 31...108..269..594..1245..2547
%t A202873 s[k_] := -1 + 2^k;
%t A202873 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 15}]];
%t A202873 L = Transpose[U]; M = L.U; TableForm[M]
%t A202873 m[i_, j_] := M[[i]][[j]];
%t A202873 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A202873 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A202873 Table[f[n], {n, 1, 12}]
%t A202873 Table[Sqrt[f[n]], {n, 1, 12}] (* A000295, Eulerian *)
%t A202873 Table[m[1, j], {j, 1, 12}]    (* A000225 *)
%t A202873 Table[m[2, j], {j, 1, 12}]    (* A053208 *)
%Y A202873 Cf. A202767.
%K A202873 nonn,tabl
%O A202873 1,2
%A A202873 _Clark Kimberling_, Dec 26 2011