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A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j). 10
1, 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000, 8537000898240926708833515201784986712482596782080000000000 (list; graph; refs; listen; history; text; internal format)
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
Equals A005249 * A000984. - Sharon Sela (sharonsela(AT)hotmail.com), Apr 18 2002
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
(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
(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
Cf. A000984, A060739. See A005249 for a formula.
Sequence in context: A099681 A362544 A062082 * A308941 A244148 A320443
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

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Last modified March 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)