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%I M4882 #78 Oct 21 2022 10:12:57
%S 1,1,12,2160,6048000,266716800000,186313420339200000,
%T 2067909047925770649600000,365356847125734485878112256000000,
%U 1028781784378569697887052962909388800000000,46206893947914691316295628839036278726983680000000000
%N Determinant of inverse Hilbert matrix.
%C a(n) = 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 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 Philip J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.
%D Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005249/b005249.txt">Table of n, a(n) for n = 0..25</a>
%H Man-Duen Choi, <a href="http://www.jstor.org/stable/2975779">Tricks or treats with the Hilbert matrix</a>, Amer. Math. Monthly, 90 (1983), 301-312.
%H Richard K. Guy, <a href="/A005249/a005249_1.pdf">Letter to N. J. A. Sloane, Sep 1986</a>.
%H John E. Lauer, <a href="/A005249/a005249.pdf">Letter to N. J. A. Sloane, Dec 1980</a>.
%H Sajad Salami, <a href="https://www.researchgate.net/publication/334263576_On_special_matrices_related_to_Cauchy_and_Toeplitz_matrices">On special matrices related to Cauchy and Toeplitz matrices</a>, Instítuto da Matemática e Estatística, Universidade Estadual do Rio de Janeiro (Brazil, 2019).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%F 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
%F The reciprocal of the determinant of an n X n matrix whose element at T(i, j) is 1/(i+j-1).
%F 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]
%F a(n) = n!*Product_{i=1..2n-1} binomial(i,floor(i/2)) = n!*|A069945(n)|. - _Peter Luschny_, Sep 18 2012
%F a(n) = Product_{i=1..2n-1} A056040(i) = A163085(2*n-1). - _Peter Luschny_, Sep 18 2012
%F 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
%F a(n) = A000178(2*n-1)/A000178(n-1)^4, for n >= 1. - _Amiram Eldar_, Oct 20 2022
%e The matrix begins:
%e 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%p with(linalg): A005249 := n-> 1/det(hilbert(n));
%t Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 0, n - 1} ]], {n, 1, 10} ]
%t Table[Denominator[Det[HilbertMatrix[n]]], {n, 0, 12}]//Quiet (* _L. Edson Jeffery_, Aug 05 2014 *)
%t Table[BarnesG[2 n + 1]/BarnesG[n + 1]^4, {n, 0, 10}] (* _Jan Mangaldan_, Sep 22 2021 *)
%o (PARI) a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)
%o (PARI) a(n)=if(n<0,0,1/matdet(mathilbert(n)))
%o (PARI) a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))
%o (J)
%o H=: % @: >: @: (+/~) @: i.
%o det=: -/ .* NB. _Roger Hui_, Oct 12 2005
%o (Sage)
%o def A005249(n):
%o swing = lambda n: factorial(n)/factorial(n//2)^2
%o return mul(swing(i) for i in (1..2*n-1))
%o [A005249(i) for i in (0..10)] # _Peter Luschny_, Sep 18 2012
%o (GAP) List([0..10],n->Product([1..n-1],k->(2*k+1)*Binomial(2*k,k)^2)); # _Muniru A Asiru_, Jul 07 2018
%Y Cf. A000178, A000515, A067689, A060739, A069945, A056040, A163085, A074962.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E 1 more term from _Jud McCranie_, Jul 16 2000
%E Additional comments from _Robert G. Wilson v_, Feb 06 2002
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