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

0,2

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]

Links

Table of n, a(n) for n=0..59.

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

Cf. A003983, A202678.

Sequence in context: A252836 A286101 A072454 * A208637 A341867 A252938

Adjacent sequences: A115213 A115214 A115215 * A115217 A115218 A115219

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 16 2006

Status

approved