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A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).

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