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%I #35 Nov 24 2023 18:39:33
%S 1,2,72,43200,423360000,67212633600000,172153600393420800000,
%T 7097063852481244869427200000,4702142622508202833251304734720000000,
%U 50019370356486058711268515056654483456000000000,8537000898240926708833515201784986712482596782080000000000
%N Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).
%D Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
%D 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.
%H Alois P. Heinz, <a href="/A067689/b067689.txt">Table of n, a(n) for n = 0..40</a> (terms n = 1..25 from T. D. Noe)
%F Equals A005249 * A000984. - Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002
%F a(n) = A163085(2*n). - _Peter Luschny_, Sep 18 2012
%F 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
%F 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
%e The matrix begins:
%e 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
%e 1/3 1/4 1/5 1/6 1/7 1/8 1/9 ...
%e 1/4 1/5 1/6 1/7 1/8 1/9 1/10 ...
%e 1/5 1/6 1/7 1/8 1/9 1/10 1/11 ...
%e 1/6 1/7 1/8 1/9 1/10 1/11 1/12 ...
%e 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
%p a:= n-> 1/LinearAlgebra[Determinant](Matrix(n, (i,j)-> 1/(i+j))):
%p seq(a(n), n=0..11); # _Alois P. Heinz_, Nov 24 2023
%t Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 1, n} ]], {n, 1, 10} ]
%t 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_ *)
%o (Sage)
%o def A067689(n):
%o swing = lambda n: factorial(n)/factorial(n//2)^2
%o return mul(swing(i) for i in (0..2*n))
%o [A067689(i) for i in (1..9)] # _Peter Luschny_, Sep 18 2012
%o (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
%Y Cf. A000984, A060739. See A005249 for a formula.
%K nonn,nice
%O 0,2
%A _Robert G. Wilson v_, Feb 04 2002
%E a(0)=1 prepended by _Alois P. Heinz_, Nov 24 2023
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