A005249 Determinant of inverse Hilbert matrix.
1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000
Offset: 0
Examples
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 ...
References
- 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).
Links
- 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.
Programs
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GAP
List([0..10],n->Product([1..n-1],k->(2*k+1)*Binomial(2*k,k)^2)); # Muniru A Asiru, Jul 07 2018
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J
H=: % @: >: @: (+/~) @: i. det=: -/ .* NB. Roger Hui, Oct 12 2005
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Maple
with(linalg): A005249 := n-> 1/det(hilbert(n));
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Mathematica
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 *) -
PARI
a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)
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PARI
a(n)=if(n<0,0,1/matdet(mathilbert(n)))
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PARI
a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))
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Sage
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
Formula
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) ~ 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
Extensions
1 more term from Jud McCranie, Jul 16 2000
Additional comments from Robert G. Wilson v, Feb 06 2002
Comments