%I A005249 M4882
%S A005249 1,1,12,2160,6048000,266716800000,186313420339200000,2067909047925770649600000,
%T A005249 365356847125734485878112256000000,1028781784378569697887052962909388800000000,
%U A005249 46206893947914691316295628839036278726983680000000000
%N A005249 Determinant of inverse Hilbert matrix.
%C A005249 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).
%C A005249 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
%D A005249 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005249 M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 
               90 (1983), 301-312.
%D A005249 Jerry Glynn & Theodore Gray, "The Beginner's Guide to Mathematica Version 
               4," Cambridge University Press, Cambridge UK, 2000, page 76.
%D A005249 P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, 
               p. 288.
%H A005249 T. D. Noe, <a href="b005249.txt">Table of n, a(n) for n=0..25</a>
%H A005249 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HilbertMatrix.html">Link to a section of The World of Mathematics.</
               a>
%F A005249 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
%F A005249 The reciprocal of the determinant of an n X n matrix whose element at 
               T(i, j) is 1/(i+j-1).
%e A005249 The matrix begins:
%e A005249 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e A005249 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e A005249 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e A005249 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e A005249 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e A005249 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%p A005249 with(linalg): A005249 := n-> 1/det(hilbert(n));
%t A005249 Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 0, n - 1} ]], {n, 
               1, 10} ]
%o A005249 (PARI) a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)
%o A005249 (PARI) a(n)=if(n<0,0,1/matdet(mathilbert(n)))
%o A005249 (PARI) a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))
%o A005249 (J programming language, http://www.jsoftware.com) - from Roger Hui (RHui000(AT)shaw.ca), 
               Oct 12 2005:
%o A005249 H=: % @: >: @: (+/~) @: i.
%o A005249 det=: -/ .*
%Y A005249 Cf. A000515, A067689, A060739.
%Y A005249 Sequence in context: A012675 A101812 A064074 this_sequence A129206 A135398 
               A010053
%Y A005249 Adjacent sequences: A005246 A005247 A005248 this_sequence A005250 A005251 
               A005252
%K A005249 nonn,easy,nice
%O A005249 0,3
%A A005249 N. J. A. Sloane (njas(AT)research.att.com).
%E A005249 1 more term from Jud McCranie (j.mccranie(AT)comcast.net), Jul 16 2000
%E A005249 Additional comments from rgwv, Feb 06, 2002.