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, 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

Author

Jianing Song, Oct 18 2021

Keywords

• nonn
• tabl

Examples

			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,
  ...
		

Links

• Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 rows)
• Eric Weisstein's World of Mathematics, Hilbert Matrix

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).

Programs

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 *)

PARI

T(n,k) = (1/mathilbert(n))[k,k]