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A115216 "Correlation triangle" for 2^n. 3

%I

%S 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,

%T 84,80,64,128,160,168,170,170,168,160,128,256,320,336,340,341,340,336,

%U 320,256,512,640,672,680,682,682,680,672,640,512,1024,1280,1344,1360,1364

%N "Correlation triangle" for 2^n.

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

%C Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.

%C 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]

%F T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).

%F G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - _Christian G. Bower_, Jan 17 2006

%e Triangle begins

%e 1,

%e 2, 2,

%e 4, 5, 4,

%e 8, 10, 10, 8,

%e 16, 20, 21, 20, 16,

%e 32, 40, 42, 42, 40, 32,

%e ...

%e Northwest corner of square matrix:

%e 1....2....4....8....16

%e 2....5....10...20...40

%e 4....10...21...42...85

%e 8....20...41...85...170

%e 16...40...84...170..341

%e ..

%t (* A115216 as a square matrix *)

%t s[k_] := 2^(k - 1);

%t U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];

%t L = Transpose[U]; M = L.U; TableForm[M]

%t m[i_, j_] := M[[i]][[j]];

%t Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]

%t f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]

%t Table[f[n], {n, 1, 12}]

%t Table[Sqrt[f[n]], {n, 1, 12}] (* -1+2^n *)

%t Table[m[n, n], {n, 1, 12}] (* A002450 *)

%t (* _Clark Kimberling_, Dec 26 2011 *)

%Y Cf. A003983, A202678.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Jan 16 2006

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Last modified June 13 10:36 EDT 2021. Contains 344988 sequences. (Running on oeis4.)