OFFSET
0,2
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)
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
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 ...
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
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
(SageMath)
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
KEYWORD
nonn,nice
AUTHOR
Robert G. Wilson v, Feb 04 2002
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Nov 24 2023
STATUS
approved