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A115216 "Correlation triangle" for 2^n.

%I A115216 #20 Feb 27 2018 23:19:22
%S A115216 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 A115216 84,80,64,128,160,168,170,170,168,160,128,256,320,336,340,341,340,336,
%U A115216 320,256,512,640,672,680,682,682,680,672,640,512,1024,1280,1344,1360,1364
%N A115216 "Correlation triangle" for 2^n.
%C A115216 Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217.
%C A115216 Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
%C A115216 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 A115216 T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).
%F A115216 G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - _Christian G. Bower_, Jan 17 2006
%e A115216 Triangle begins
%e A115216   1,
%e A115216   2, 2,
%e A115216   4, 5, 4,
%e A115216   8, 10, 10, 8,
%e A115216   16, 20, 21, 20, 16,
%e A115216   32, 40, 42, 42, 40, 32,
%e A115216   ...
%e A115216 Northwest corner of square matrix:
%e A115216   1....2....4....8....16
%e A115216   2....5....10...20...40
%e A115216   4....10...21...42...85
%e A115216   8....20...41...85...170
%e A115216   16...40...84...170..341
%e A115216   ..
%t A115216 (* A115216 as a square matrix *)
%t A115216 s[k_] := 2^(k - 1);
%t A115216 U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];
%t A115216 L = Transpose[U]; M = L.U; TableForm[M]
%t A115216 m[i_, j_] := M[[i]][[j]];
%t A115216 Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
%t A115216 f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
%t A115216 Table[f[n], {n, 1, 12}]
%t A115216 Table[Sqrt[f[n]], {n, 1, 12}]  (* -1+2^n *)
%t A115216 Table[m[n, n], {n, 1, 12}]  (* A002450 *)
%t A115216 (* _Clark Kimberling_, Dec 26 2011 *)
%Y A115216 Cf. A003983, A202678.
%K A115216 easy,nonn,tabl
%O A115216 0,2
%A A115216 _Paul Barry_, Jan 16 2006