|
| |
|
|
A005249
|
|
Determinant of inverse Hilbert matrix.
(Formerly M4882)
|
|
26
| |
|
|
1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000
(list; graph; refs; listen; history; 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
| M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
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
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
|
|
|
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 (benoit7848c(AT)orange.fr), 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 [From Enrique Perez Herrero (psychgeometry(AT)gmail.com), Mar 31 2010]
|
|
|
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} ]
|
|
|
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 (RHui000(AT)shaw.ca), Oct 12 2005:
H=: % @: >: @: (+/~) @: i.
det=: -/ .*
|
|
|
CROSSREFS
| Cf. A000515, A067689, A060739.
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 (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| 1 more term from Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Jul 16 2000
Additional comments from rgwv, Feb 06, 2002.
|
| |
|
|
