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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.
1
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
1,2
LINKS
Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 rows)
Eric Weisstein's World of Mathematics, Hilbert Matrix
EXAMPLE
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,
...
MAPLE
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
MATHEMATICA
T[n_, k_] := Inverse[HilbertMatrix[n]][[k, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 18 2021 *)
PROG
(PARI) T(n, k) = (1/mathilbert(n))[k, k]
CROSSREFS
Cf. A189766 (row sums), A189765, A005249.
A210356 gives the maximum value of each row and A210357 gives the positions of the maximum values.
Main diagonal gives A000515(n-1).
Sequence in context: A307853 A334768 A247327 * A238581 A063608 A074258
KEYWORD
nonn,tabl
AUTHOR
Jianing Song, Oct 18 2021
STATUS
approved