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A189765
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Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.
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5
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1, 4, -6, 12, 9, -36, 192, 30, -180, 180, 16, -120, 1200, 240, -2700, 6480, -140, 1680, -4200, 2800, 25, -300, 4800, 1050, -18900, 79380, -1400, 26880, -117600, 179200, 630, -12600, 56700, -88200, 44100, 36, -630, 14700, 3360, -88200, 564480, -7560, 211680
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OFFSET
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1,2
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COMMENTS
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The n-th order Hilbert matrix has elements h(i,j) = 1/(i+j-1) for 1 <= i,j <=n. Only the lower triangular matrix is shown because the Hilbert matrix and its inverse are symmetric. The n-th row begins with n^2 and ends with A000515(n+1).
The sums of select rows of the inverse matrix are sequences A002457, A002736, A002738, A007531, and A054559.
The largest magnitude in the matrix is A210356(n). - T. D. Noe, Mar 28 2012
The sum of the elements of the n-th matrix is n^2. - T. D. Noe, Apr 02 2012
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LINKS
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T. D. Noe, Rows n = 1..25, flattened
Eric W. Weisstein, MathWorld: Hilbert Matrix
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FORMULA
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a(n,i,j) = (-1)^(i+j) (i+j-1) binomial(n+i-1, n-j) binomial(n+j-1, n-i) binomial(i+j-2, i-1)^2 is the (i,j) element of the inverse of the n-th Hilbert matrix.
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EXAMPLE
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Row 3 is 9, -36, 192, 30, -180, 180 which corresponds to the inverse
9 -36 30
-36 192 -180
30 -180 180
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MATHEMATICA
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lowerTri[m_List] := Module[{n = Length[m]}, Flatten[Table[Take[m[[i]], i], {i, n}]]]; Flatten[Table[lowerTri[Inverse[HilbertMatrix[n]]], {n, 6}]]
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CROSSREFS
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Cf. A002457, A002736, A002738, A005249 (determinant), A007531, A054559, A189766 (trace).
Sequence in context: A272063 A152678 A110758 * A074162 A038040 A143356
Adjacent sequences: A189762 A189763 A189764 * A189766 A189767 A189768
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KEYWORD
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sign,tabf
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AUTHOR
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T. D. Noe, May 02 2011
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STATUS
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approved
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