|
Search: id:A005249
|
|
|
| A005249 |
|
Determinant of inverse Hilbert matrix.
(Formerly M4882)
|
|
+0
23
|
|
| 1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
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
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
Jerry Glynn & Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.
|
|
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).
|
|
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} ]
|
|
PROGRAM
|
(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: A012675 A101812 A064074 this_sequence A129206 A135398 A010053
Adjacent sequences: A005246 A005247 A005248 this_sequence A005250 A005251 A005252
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
1 more term from Jud McCranie (j.mccranie(AT)comcast.net), Jul 16 2000
Additional comments from rgwv, Feb 06, 2002.
|
|
|
Search completed in 0.002 seconds
|
