A210356
Maximum modulus in the inverse of Hilbert's matrix.
Original entry on oeis.org
1, 12, 192, 6480, 179200, 4410000, 133402500, 4249941696, 122367445200, 3480673996800, 117643011932160, 3659449159080000, 106518477825760000, 3521767173114190000, 114708987924290760000, 3525270042097046880000, 110552468520163390156800
Offset: 1
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Table[im = Inverse[HilbertMatrix[n]]; Max[Abs[Flatten[im]]], {n, 20}]
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for(n=1,17, my(h=1/mathilbert(n),s=0); for(j=1,n, for(k=1,n, s=max(s,h[j,k]))); print1(s,", ")) \\ Hugo Pfoertner, Feb 11 2020
A189766
Trace of the inverse of the n-th order Hilbert matrix.
Original entry on oeis.org
1, 16, 381, 10496, 307505, 9316560, 288307285, 9052917760, 287307428985, 9192433560080, 295998598024613, 9580548525151488, 311414673789269713, 10158681128480830288, 332394269045633574405, 10904463909222273843200, 358543696456299951516425
Offset: 1
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Table[Trace[Inverse[HilbertMatrix[n]]], {n, 20}] (* or *)
Table[n^2 HypergeometricPFQ[{1/2, 1-n, 1-n, 1+n, 1+n}, {1, 1, 1, 3/2}, 1], {n, 20}]
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a(n) = trace(1/mathilbert(n)) \\ Jianing Song, Oct 18 2021
A210358
Minimum value in the inverse of Hilbert's matrix.
Original entry on oeis.org
1, -6, -180, -4200, -117600, -3969000, -115259760, -3030051024, -106051785840, -3363975014400, -98227319533200, -3110531785218000, -103492384705710000, -3211851661880141280, -96355549856404238400, -3303308595002047632000, -106681852579497947388000
Offset: 1
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Table[im = Inverse[HilbertMatrix[n]]; Min[Flatten[im]], {n, 20}]
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a(n)=vecmin(1/mathilbert(n)) \\ Hugo Pfoertner, Feb 11 2020
A348419
Triangular table read by rows: T(n,k) is the k-th entry of the main diagonal of the inverse Hilbert matrix of order n.
Original entry on oeis.org
1, 4, 12, 9, 192, 180, 16, 1200, 6480, 2800, 25, 4800, 79380, 179200, 44100, 36, 14700, 564480, 3628800, 4410000, 698544, 49, 37632, 2857680, 40320000, 133402500, 100590336, 11099088, 64, 84672, 11430720, 304920000, 2134440000, 4249941696, 2175421248, 176679360
Offset: 1
The inverse Hilbert matrix of order 4 is given by
[ 16 -120 240 -140]
[-120 1200 -2700 1680]
[ 240 -2700 6480 -4200]
[-140 1680 -4200 2800].
Hence the 4th row is 16, 1200, 6480, 2800.
The first 8 rows of the table are:
1,
4, 12,
9, 192, 180,
16, 1200, 6480, 2800,
25, 4800, 79380, 179200, 44100,
36, 14700, 564480, 3628800, 4410000, 698544,
49, 37632, 2857680, 40320000, 133402500, 100590336, 11099088,
64, 84672, 11430720, 304920000, 2134440000, 4249941696, 2175421248, 176679360,
...
A210356 gives the maximum value of each row and
A210357 gives the positions of the maximum values.
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T:= n-> (M-> seq(M[i, i], i=1..n))(1/LinearAlgebra[HilbertMatrix](n)):
seq(T(n), n=1..8); # Alois P. Heinz, Jun 19 2022
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T[n_, k_] := Inverse[HilbertMatrix[n]][[k, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 18 2021 *)
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T(n,k) = (1/mathilbert(n))[k,k]
Showing 1-4 of 4 results.
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