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A067689 Inverse of determinant of n X n matrix whose (i,j)-th element is 1/(i+j). 10
2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

T. D. Noe, Table of n, a(n) for n = 1..25

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 ...

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: A317346 A099681 A062082 * A308941 A244148 A320443

Adjacent sequences:  A067686 A067687 A067688 * A067690 A067691 A067692

KEYWORD

nonn,nice

AUTHOR

Robert G. Wilson v, Feb 04 2002

STATUS

approved

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)