A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j).
1, 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000, 8537000898240926708833515201784986712482596782080000000000
Offset: 0
Examples
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 ...
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
- Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..25 from T. D. Noe)
Programs
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Maple
a:= n-> 1/LinearAlgebra[Determinant](Matrix(n, (i,j)-> 1/(i+j))): seq(a(n), n=0..11); # Alois P. Heinz, Nov 24 2023
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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 *) -
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
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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
Formula
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
Extensions
a(0)=1 prepended by Alois P. Heinz, Nov 24 2023