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 A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j). 10
 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 REFERENCES Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76. G. Pólya and G. Szegő, Aufgaben und Lehrsätze aus der Analysis II, Vierte Auflage, Heidelberger Taschenbücher, Springer, 1971, p. 98, 3. and p. 299, 3. LINKS T. D. Noe, Table of n, a(n) for n = 1..25 FORMULA Equals A005249 * A000984. - Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002 a(n) = A163085(2*n). - Peter Luschny, Sep 18 2012 a(n) ~ A^3 * 2^(2*n^2 + n - 1/12) / (exp(1/4) * n^(1/4) * Pi^(n+1/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015 a(n) = Prod_{i=1..n}(Prod_{j=1..n} (i+j)) / Prod_{i=1..n}(Prod_{j=1..n-1} (i-j)^2),  n >= 1. See the Pólya and Szegő reference (special case) with the original Cauchy reference. - Wolfdieter Lang, Apr 25 2016 EXAMPLE The matrix begins: 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ... 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ... 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ... 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ... 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ... 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ... MATHEMATICA Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 1, n} ]], {n, 1, 10} ] a[n_] := Product[ k!/Quotient[k, 2]!^2, {k, 0, 2*n}]; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, Oct 17 2013, after Peter Luschny *) PROG (Sage) def A067689(n):     swing = lambda n: factorial(n)/factorial(n//2)^2     return mul(swing(i) for i in (0..2*n)) [A067689(i) for i in (1..9)] # Peter Luschny, Sep 18 2012 (PARI) a(n)=prod(k=0, n-1, (2*k)!*(2*k+1)!/k!^4)*binomial(2*n, n) \\ Charles R Greathouse IV, Feb 07 2017 CROSSREFS Cf. A000984, A060739. See A005249 for a formula. Sequence in context: A317346 A099681 A062082 * A244148 A079478 A221709 Adjacent sequences:  A067686 A067687 A067688 * A067690 A067691 A067692 KEYWORD nonn,nice AUTHOR Robert G. Wilson v, Feb 04 2002 STATUS approved

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Last modified October 16 08:03 EDT 2018. Contains 316259 sequences. (Running on oeis4.)