login
A115216
"Correlation triangle" for 2^n.
3
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]
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
Sequence in context: A252836 A286101 A072454 * A208637 A341867 A252938
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Jan 16 2006
STATUS
approved