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A067689
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Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).
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10
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1, 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000, 8537000898240926708833515201784986712482596782080000000000
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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FORMULA
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Equals A005249 * A000984. - Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002
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
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EXAMPLE
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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 ...
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MAPLE
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a:= n-> 1/LinearAlgebra[Determinant](Matrix(n, (i, j)-> 1/(i+j))):
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MATHEMATICA
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Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 1, n} ]], {n, 1, 10} ]
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PROG
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(Sage)
swing = lambda n: factorial(n)/factorial(n//2)^2
return mul(swing(i) for i in (0..2*n))
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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