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 A005249 Determinant of inverse Hilbert matrix. (Formerly M4882) 37
 1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). 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 Philip J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288. Jerry Glynn and 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 Man-Duen Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312. Richard K. Guy, Letter to N. J. A. Sloane, Sep 1986. John E. Lauer, Letter to N. J. A. Sloane, Dec 1980. Sajad Salami, On special matrices related to Cauchy and Toeplitz matrices, Instítuto da Matemática e Estatística, Universidade Estadual do Rio de Janeiro (Brazil, 2019). Eric Weisstein's World of Mathematics, Hilbert Matrix. FORMULA 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 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)*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] a(n) = n!*Product_{i=1..2n-1} binomial(i,floor(i/2)) = n!*|A069945(n)|. - Peter Luschny, Sep 18 2012 a(n) = Product_{i=1..2n-1} A056040(i) = A163085(2*n-1). - Peter Luschny, Sep 18 2012 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 a(n) = A000178(2*n-1)/A000178(n-1)^4, for n >= 1. - Amiram Eldar, Oct 20 2022 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 *) Table[BarnesG[2 n + 1]/BarnesG[n + 1]^4, {n, 0, 10}] (* Jan Mangaldan, Sep 22 2021 *) 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) H=: % @: >: @: (+/~) @: i. det=: -/ .* NB. Roger Hui, Oct 12 2005 (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 (GAP) List([0..10], n->Product([1..n-1], k->(2*k+1)*Binomial(2*k, k)^2)); # Muniru A Asiru, Jul 07 2018 CROSSREFS Cf. A000178, A000515, A067689, A060739, A069945, A056040, A163085, A074962. 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 December 11 01:27 EST 2023. Contains 367717 sequences. (Running on oeis4.)