A060739 -id:A060739 - cp's OEIS frontend

A005249 Determinant of inverse Hilbert matrix.

1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000

Offset: 0

N. J. A. Sloane

a(n) = 1/determinant of M(n)*(-1)^floor(n/2) where M(n) is the n X n matrix m(i,j)=1/(i-j+n).

For n>=2, a(n) = Product k=1...(n-1) (2k+1) * C(2k,k)^2. This is a special case of the Cauchy determinant formula. A similar formula exists also for A067689. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002

			The matrix begins:
  1    1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

Philip J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.
Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..25
Man-Duen Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1986.
John E. Lauer, Letter to N. J. A. Sloane, Dec 1980.
Sajad Salami, On special matrices related to Cauchy and Toeplitz matrices, Instítuto da Matemática e Estatística, Universidade Estadual do Rio de Janeiro (Brazil, 2019).
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A000515, A067689, A060739, A069945, A056040, A163085, A074962.

List([0..10],n->Product([1..n-1],k->(2*k+1)*Binomial(2*k,k)^2)); # Muniru A Asiru, Jul 07 2018

H=: % @: >: @: (+/~) @: i.
det=: -/ .* NB. Roger Hui, Oct 12 2005

with(linalg): A005249 := n-> 1/det(hilbert(n));

Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 0, n - 1} ]], {n, 1, 10} ]
Table[Denominator[Det[HilbertMatrix[n]]], {n, 0, 12}]//Quiet (* L. Edson Jeffery, Aug 05 2014 *)
Table[BarnesG[2 n + 1]/BarnesG[n + 1]^4, {n, 0, 10}] (* Jan Mangaldan, Sep 22 2021 *)

a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)

PARI
```
a(n)=if(n<0,0,1/matdet(mathilbert(n)))
```

a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))

def A005249(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (1..2*n-1))
[A005249(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = n^n*(Product_{k=1..n-1} (n^2 - k^2)^(n-k))/Product_{k=0..n-1} k!^2. - Benoit Cloitre, Jan 15 2003
The reciprocal of the determinant of an n X n matrix whose element at T(i, j) is 1/(i+j-1).
a(n+1) = a(n)*A000515(n) = a(n)*(2*n+1)*binomial(2n,n)^2. - Enrique Pérez Herrero, Mar 31 2010 [In other words, the partial products of sequence A000515. - N. J. A. Sloane, Jul 10 2015]
a(n) = n!*Product_{i=1..2n-1} binomial(i,floor(i/2)) = n!*|A069945(n)|. - Peter Luschny, Sep 18 2012
a(n) = Product_{i=1..2n-1} A056040(i) = A163085(2*n-1). - Peter Luschny, Sep 18 2012
a(n) ~ A^3 * 2^(2*n^2 - n - 1/12) * n^(1/4) / (exp(1/4) * Pi^n), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015
a(n) = A000178(2*n-1)/A000178(n-1)^4, for n >= 1. - Amiram Eldar, Oct 20 2022

1 more term from Jud McCranie, Jul 16 2000

Additional comments from Robert G. Wilson v, Feb 06 2002

A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).

Original entry on oeis.org

1, 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000, 8537000898240926708833515201784986712482596782080000000000

Offset: 0

Robert G. Wilson v, Feb 04 2002

			The matrix begins:
1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis II, Vierte Auflage, Heidelberger Taschenbücher, Springer, 1971, p. 98, 3. and p. 299, 3.

Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..25 from T. D. Noe)

Cf. A000984, A060739. See A005249 for a formula.

a:= n-> 1/LinearAlgebra[Determinant](Matrix(n, (i,j)-> 1/(i+j))):
seq(a(n), n=0..11);  # Alois P. Heinz, Nov 24 2023

Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 1, n} ]], {n, 1, 10} ]
a[n_] := Product[ k!/Quotient[k, 2]!^2, {k, 0, 2*n}]; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, Oct 17 2013, after Peter Luschny *)

a(n)=prod(k=0, n-1, (2*k)!*(2*k+1)!/k!^4)*binomial(2*n,n) \\ Charles R Greathouse IV, Feb 07 2017

def A067689(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (0..2*n))
[A067689(i) for i in (1..9)] # Peter Luschny, Sep 18 2012

Equals A005249 * A000984. - Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002
a(n) = A163085(2*n). - Peter Luschny, Sep 18 2012
a(n) ~ A^3 * 2^(2*n^2 + n - 1/12) / (exp(1/4) * n^(1/4) * Pi^(n+1/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015
a(n) = Prod_{i=1..n}(Prod_{j=1..n} (i+j)) / Prod_{i=1..n}(Prod_{j=1..n-1} (i-j)^2),  n >= 1. See the Pólya and Szegő reference (special case) with the original Cauchy reference. - Wolfdieter Lang, Apr 25 2016

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

cp's OEIS Frontend

A005249 Determinant of inverse Hilbert matrix.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

J

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

Extensions

A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A069704 Let M_2n be the 2n X 2n matrix M_(i,j)=C(2i,j)-C(2j,i) where C(k,l) denotes the binomial coefficients; then a(n)=det(M_2n).

Views

Author

Keywords

Comments

Crossrefs