OFFSET
0,2
COMMENTS
Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
When formated as a square array, this is the self-fusion matrix (as in Example and Mathematica sections) of the sequence (2^n); for interlacing zeros of associated characteristic polynomials, see A202868. [Clark Kimberling, Dec 26 2011]
FORMULA
T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).
G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006
EXAMPLE
Triangle begins
1,
2, 2,
4, 5, 4,
8, 10, 10, 8,
16, 20, 21, 20, 16,
32, 40, 42, 42, 40, 32,
...
Northwest corner of square matrix:
1....2....4....8....16
2....5....10...20...40
4....10...21...42...85
8....20...41...85...170
16...40...84...170..341
..
MATHEMATICA
(* A115216 as a square matrix *)
s[k_] := 2^(k - 1);
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {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}]]
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}] (* -1+2^n *)
Table[m[n, n], {n, 1, 12}] (* A002450 *)
(* Clark Kimberling, Dec 26 2011 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jan 16 2006
STATUS
approved