A189766 Trace of the inverse of the n-th order Hilbert matrix.

1, 16, 381, 10496, 307505, 9316560, 288307285, 9052917760, 287307428985, 9192433560080, 295998598024613, 9580548525151488, 311414673789269713, 10158681128480830288, 332394269045633574405, 10904463909222273843200, 358543696456299951516425

Offset

1,2

Comments

See the Mathematica program for a formula in terms of a hypergeometric function.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..600

Formula

From Richard Penner, Jun 05 2011: (Start)

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.

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)

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

a(n) ~ 2^(1/4)*(17+12*sqrt(2))^n/(16*Pi^(3/2)*sqrt(n)). - Vaclav Kotesovec, Aug 18 2013

Mathematica

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}]

Prog

(PARI) a(n) = trace(1/mathilbert(n)) \\ Jianing Song, Oct 18 2021

Crossrefs

Cf. A005249 (determinant), A189765 (inverse Hilbert matrix).

Sequence in context: A201617 A235672 A235729 * A235442 A343213 A340689

Adjacent sequences: A189763 A189764 A189765 * A189767 A189768 A189769

Keyword

nonn

Author

T. D. Noe, May 02 2011

Status

approved