A115216 "Correlation triangle" for 2^n.

1, 2, 2, 4, 5, 4, 8, 10, 10, 8, 16, 20, 21, 20, 16, 32, 40, 42, 42, 40, 32, 64, 80, 84, 85, 84, 80, 64, 128, 160, 168, 170, 170, 168, 160, 128, 256, 320, 336, 340, 341, 340, 336, 320, 256, 512, 640, 672, 680, 682, 682, 680, 672, 640, 512, 1024, 1280, 1344, 1360, 1364

Offset: 0

Author

Paul Barry, Jan 16 2006

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217.

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]

Examples

			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
  ..
		

Crossrefs

Cf. A003983, A202678.

Programs

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 *)

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