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A005249 Determinant of inverse Hilbert matrix.
(Formerly M4882)
28
1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.

Jerry Glynn & 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

M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.

Eric Weisstein's World of Mathematics, Hilbert Matrix.

FORMULA

a(n)=n^n*prod(k=1, n-1, (n^2-k^2)^(n-k))/prod(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)*C(2n,n)^2. [Enrique Pérez Herrero, Mar 31 2010]

a(n) = n!*prod_{i=1..2n-1} C(i,floor(i/2)) = n!*|A069945(n)|. - Peter Luschny, Sep 18 2012

a(n) = prod_{i=1..2n-1} A056040(i) = A163085(2*n-1). - Peter Luschny, Sep 18 2012

EXAMPLE

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 ...

MAPLE

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

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 *)

PROG

(PARI) 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)))

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

(J programming language, http://www.jsoftware.com) - from Roger Hui, Oct 12 2005:

H=: % @: >: @: (+/~) @: i.

det=: -/ .*

(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

CROSSREFS

Cf. A000515, A067689, A060739, A069945.

Sequence in context: A175014 A101812 A064074 * A177069 A204681 A205157

Adjacent sequences:  A005246 A005247 A005248 * A005250 A005251 A005252

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

1 more term from Jud McCranie, Jul 16 2000

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

STATUS

approved

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Last modified October 24 05:24 EDT 2014. Contains 248500 sequences.