This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A189766 #31 Oct 18 2021 08:10:08
%S A189766 1,16,381,10496,307505,9316560,288307285,9052917760,287307428985,
%T A189766 9192433560080,295998598024613,9580548525151488,311414673789269713,
%U A189766 10158681128480830288,332394269045633574405,10904463909222273843200,358543696456299951516425
%N A189766 Trace of the inverse of the n-th order Hilbert matrix.
%C A189766 See the Mathematica program for a formula in terms of a hypergeometric function.
%H A189766 Vincenzo Librandi, <a href="/A189766/b189766.txt">Table of n, a(n) for n = 1..600</a>
%F A189766 From _Richard Penner_, Jun 05 2011: (Start)
%F A189766 a(n) = n * A178790(n) = Sum_{k=0..n-1} (2*k+1)*binomial(n+k, 2*k+1)^2 * binomial(2*k,k)^2.
%F A189766 a(n) = Sum_{k=1..n} A005408(k)*A005259(k-1) = Sum_{k=0..n-1} (2*k+1) * Sum_{j=0..k} binomial(k+j,j)^2 * binomial(k,j)^2. (End)
%F A189766 Recurrence: (n-1)^3*(2*n-5)*a(n) = (2*n-5)*(35*n^3 - 122*n^2 + 132*n - 40)*a(n-1) - (2*n-1)*(35*n^3 - 193*n^2 + 345*n - 203)*a(n-2) + (n-2)^3*(2*n-1)*a(n-3). - _Vaclav Kotesovec_, Aug 18 2013
%F A189766 a(n) ~ 2^(1/4)*(17+12*sqrt(2))^n/(16*Pi^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, Aug 18 2013
%t A189766 Table[Trace[Inverse[HilbertMatrix[n]]], {n, 20}] (* or *)
%t A189766 Table[n^2 HypergeometricPFQ[{1/2, 1-n, 1-n, 1+n, 1+n}, {1, 1, 1, 3/2}, 1], {n, 20}]
%o A189766 (PARI) a(n) = trace(1/mathilbert(n)) \\ _Jianing Song_, Oct 18 2021
%Y A189766 Cf. A005249 (determinant), A189765 (inverse Hilbert matrix).
%K A189766 nonn
%O A189766 1,2
%A A189766 _T. D. Noe_, May 02 2011