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 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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 A252938 A229402 Adjacent sequences:  A115213 A115214 A115215 * A115217 A115218 A115219 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Jan 16 2006 STATUS approved

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Last modified June 24 13:17 EDT 2019. Contains 324325 sequences. (Running on oeis4.)