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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.

%I A348419 #27 Feb 16 2025 08:34:02
%S A348419 1,4,12,9,192,180,16,1200,6480,2800,25,4800,79380,179200,44100,36,
%T A348419 14700,564480,3628800,4410000,698544,49,37632,2857680,40320000,
%U A348419 133402500,100590336,11099088,64,84672,11430720,304920000,2134440000,4249941696,2175421248,176679360
%N 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.
%H A348419 Jianing Song, <a href="/A348419/b348419.txt">Table of n, a(n) for n = 1..5050</a> (first 100 rows)
%H A348419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>
%e A348419 The inverse Hilbert matrix of order 4 is given by
%e A348419   [  16  -120   240  -140]
%e A348419   [-120  1200 -2700  1680]
%e A348419   [ 240 -2700  6480 -4200]
%e A348419   [-140  1680 -4200  2800].
%e A348419 Hence the 4th row is 16, 1200, 6480, 2800.
%e A348419 The first 8 rows of the table are:
%e A348419   1,
%e A348419   4, 12,
%e A348419   9, 192, 180,
%e A348419   16, 1200, 6480, 2800,
%e A348419   25, 4800, 79380, 179200, 44100,
%e A348419   36, 14700, 564480, 3628800, 4410000, 698544,
%e A348419   49, 37632, 2857680, 40320000, 133402500, 100590336, 11099088,
%e A348419   64, 84672, 11430720, 304920000, 2134440000, 4249941696, 2175421248, 176679360,
%e A348419   ...
%p A348419 T:= n-> (M-> seq(M[i, i], i=1..n))(1/LinearAlgebra[HilbertMatrix](n)):
%p A348419 seq(T(n), n=1..8);  # _Alois P. Heinz_, Jun 19 2022
%t A348419 T[n_, k_] := Inverse[HilbertMatrix[n]][[k, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Oct 18 2021 *)
%o A348419 (PARI) T(n,k) = (1/mathilbert(n))[k,k]
%Y A348419 Cf. A189766 (row sums), A189765, A005249.
%Y A348419 A210356 gives the maximum value of each row and A210357 gives the positions of the maximum values.
%Y A348419 Main diagonal gives A000515(n-1).
%K A348419 nonn,tabl
%O A348419 1,2
%A A348419 _Jianing Song_, Oct 18 2021