A163085 -id:A163085 - cp's OEIS frontend

A163086 Product of first n terms of A163085.

1, 1, 2, 24, 1728, 3732480, 161243136000, 975198486528000000, 412860031256494080000000000, 110116706384632080236544000000000000000, 7401233839469056679744633202278400000000000000000000

Offset: 0

Peter Luschny, Jul 21 2009

nonn

Cf. A056040, A163085, A055462, A000178.

a := proc(n) local i; mul(A163085(i),i=0..n) end;

b[0] = 1; b[n_] := b[n] = b[n-1] n! / Floor[n/2]!^2;
a[n_] := Product[b[k], {k, 0, n}];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 11 2019 *)

def A163086(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i+1)^(n-i) for i in (0..n))
[A163086(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = product_{i=0..n} A056040(i+1)^(n-i). - Peter Luschny, Sep 18 2012

A216628 a(n) = A163085(n)/n!.

Original entry on oeis.org

1, 1, 1, 2, 3, 18, 60, 1200, 10500, 735000, 18522000, 4667544000, 359400888000, 332086420512000, 81408613942656000, 279394363051195392000, 224737840779305293440000, 2892376010829659126572800000, 7812628980363223707442752000000, 379850021025259936655866602240000000

Offset: 0

Peter Luschny, Sep 18 2012

nonn

Cf. A056040.

def A216628(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    A163085 = lambda n: mul(swing(i) for i in (0..n))
    return A163085(n)/factorial(n)
[A216628(n) for n in (0..19)]

A216629 a(n) = A163085(n)/(floor(n/2)!)^2.

Original entry on oeis.org

1, 1, 2, 12, 18, 540, 1200, 168000, 735000, 463050000, 4667544000, 12938431968000, 332086420512000, 3989022083190144000, 279394363051195392000, 14383221809875538780160000, 2892376010829659126572800000, 632822947409421120302862912000000

Offset: 0

Peter Luschny, Sep 18 2012

nonn

Cf. A069651.

def A216629(n): return A163085(n)/factorial(n//2)^2
[A216629(n) for n in (0..17)]

A005249 Determinant of inverse Hilbert matrix.

Original entry on oeis.org

1, 1, 12, 2160, 6048000, 266716800000, 186313420339200000, 2067909047925770649600000, 365356847125734485878112256000000, 1028781784378569697887052962909388800000000, 46206893947914691316295628839036278726983680000000000

Offset: 0

N. J. A. Sloane

a(n) = 1/determinant of M(n)*(-1)^floor(n/2) where M(n) is the n X n matrix m(i,j)=1/(i-j+n).

For n>=2, a(n) = Product k=1...(n-1) (2k+1) * C(2k,k)^2. This is a special case of the Cauchy determinant formula. A similar formula exists also for A067689. - Sharon Sela (sharonsela(AT)hotmail.com), Mar 23 2002

			The matrix begins:
  1    1/2  1/3  1/4  1/5  1/6  1/7  1/8  ...
  1/2  1/3  1/4  1/5  1/6  1/7  1/8  1/9  ...
  1/3  1/4  1/5  1/6  1/7  1/8  1/9  1/10 ...
  1/4  1/5  1/6  1/7  1/8  1/9  1/10 1/11 ...
  1/5  1/6  1/7  1/8  1/9  1/10 1/11 1/12 ...
  1/6  1/7  1/8  1/9  1/10 1/11 1/12 1/13 ...

Philip J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 288.
Jerry Glynn and Theodore Gray, "The Beginner's Guide to Mathematica Version 4," Cambridge University Press, Cambridge UK, 2000, page 76.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..25
Man-Duen Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.
Richard K. Guy, Letter to N. J. A. Sloane, Sep 1986.
John E. Lauer, Letter to N. J. A. Sloane, Dec 1980.
Sajad Salami, On special matrices related to Cauchy and Toeplitz matrices, Instítuto da Matemática e Estatística, Universidade Estadual do Rio de Janeiro (Brazil, 2019).
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A000515, A067689, A060739, A069945, A056040, A163085, A074962.

List([0..10],n->Product([1..n-1],k->(2*k+1)*Binomial(2*k,k)^2)); # Muniru A Asiru, Jul 07 2018

H=: % @: >: @: (+/~) @: i.
det=: -/ .* NB. Roger Hui, Oct 12 2005

with(linalg): A005249 := n-> 1/det(hilbert(n));

Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 0, n - 1} ]], {n, 1, 10} ]
Table[Denominator[Det[HilbertMatrix[n]]], {n, 0, 12}]//Quiet (* L. Edson Jeffery, Aug 05 2014 *)
Table[BarnesG[2 n + 1]/BarnesG[n + 1]^4, {n, 0, 10}] (* Jan Mangaldan, Sep 22 2021 *)

a(n)=n^n*prod(k=1,n-1,(n^2-k^2)^(n-k))/prod(k=0,n-1,k!^2)

PARI
```
a(n)=if(n<0,0,1/matdet(mathilbert(n)))
```

a(n)=if(n<0,0,prod(k=0,n-1,(2*k)!*(2*k+1)!/k!^4))

def A005249(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (1..2*n-1))
[A005249(i) for i in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = n^n*(Product_{k=1..n-1} (n^2 - k^2)^(n-k))/Product_{k=0..n-1} k!^2. - Benoit Cloitre, Jan 15 2003
The reciprocal of the determinant of an n X n matrix whose element at T(i, j) is 1/(i+j-1).
a(n+1) = a(n)*A000515(n) = a(n)*(2*n+1)*binomial(2n,n)^2. - Enrique Pérez Herrero, Mar 31 2010 [In other words, the partial products of sequence A000515. - N. J. A. Sloane, Jul 10 2015]
a(n) = n!*Product_{i=1..2n-1} binomial(i,floor(i/2)) = n!*|A069945(n)|. - Peter Luschny, Sep 18 2012
a(n) = Product_{i=1..2n-1} A056040(i) = A163085(2*n-1). - Peter Luschny, Sep 18 2012
a(n) ~ A^3 * 2^(2*n^2 - n - 1/12) * n^(1/4) / (exp(1/4) * Pi^n), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, May 01 2015
a(n) = A000178(2*n-1)/A000178(n-1)^4, for n >= 1. - Amiram Eldar, Oct 20 2022

1 more term from Jud McCranie, Jul 16 2000

Additional comments from Robert G. Wilson v, Feb 06 2002

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

Original entry on oeis.org

1, 2, 72, 43200, 423360000, 67212633600000, 172153600393420800000, 7097063852481244869427200000, 4702142622508202833251304734720000000, 50019370356486058711268515056654483456000000000, 8537000898240926708833515201784986712482596782080000000000

Offset: 0

Robert G. Wilson v, Feb 04 2002

			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 ...

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.

Alois P. Heinz, Table of n, a(n) for n = 0..40 (terms n = 1..25 from T. D. Noe)

Cf. A000984, A060739. See A005249 for a formula.

a:= n-> 1/LinearAlgebra[Determinant](Matrix(n, (i,j)-> 1/(i+j))):
seq(a(n), n=0..11);  # Alois P. Heinz, Nov 24 2023

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 *)

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

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

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

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

A060739 a(n) = (-1)^(n(n-1)/2) * Product_{k=0,...,n-1} (n+k-1)!/((k!)^2 * (n-1-k)!).

Original entry on oeis.org

1, 1, -2, -36, 7200, 17640000, -560105280000, -239102222768640000, 1408147589778024775680000, 116620600756651855983415296000000, -137839975629646325813680872620851200000000, -2352568589682795058651211199786427114330521600000000

Offset: 0

Noam Katz (noamkj(AT)hotmail.com), Apr 25 2001

Let A_n be the matrix of size n X n defined by: A_n[i,j] = 1/(binomial coefficient i+j-2 over i-1) = 1/C(i+j-2,i-1) where 1 <= i,j <= n. The diagonals of this matrix are the reciprocals of the entries in the Pascal triangle. Then a(n) = 1/det(A_n) = det((A_n)^(-1)).
From the formula for a(n) it follows that the determinant of (A_n)^(-1) is an integer. By inspecting the values of (A_n)^(-1) for small values of n it looks like (A_n)^(-1) is actually a matrix of integers but I do not have a proof of this fact.
Let M_n be the n X n matrix with M_n(i,j)=i/(i+j); then |a(n-1)|=1/det(M_n). - Benoit Cloitre, Apr 21 2002
Also related to the multinomial coefficients (i+j)!/i!/j! : abs(a(n))=(1/detQ_n-1) where Q_n is the n X n matrix q(i,j)=i!j!/(i+j)! - Benoit Cloitre, May 30 2002
From Alexander Adamchuk, Nov 14 2009: (Start)
Also a(n) = (-1)^(n(n-1)/2) * Product[ Binomial[2k,k]^2/2, {k,1,n-1} ].
It is simpler definition of a(n).
It follows from the observation that Sqrt[ Abs[ a(n+1)/a(n)/2 ] ] = {1, 3, 10, 35, 126, 462, ...} = C(2n+1, n+1) = A001700. (End)

			Here is the matrix A_4 for n=4: [1, 1, 1, 1; 1, 1/2, 1/3, 1/4; 1, 1/3, 1/6, 1/10; 1, 1/4, 1/10, 1/20]; a(4) = 7200 because det(A_4) = 1/7200

Harry J. Smith, Table of n, a(n) for n=0,...,43
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv preprint arXiv:1405.6315 [math.CO], 2014.
Doron Zeilberger, Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico, arXiv:1405.6315 [math.CO], 2014.
Doron Zeilberger, Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico, Amer. Math. Monthly 103 (1996), 501-502.

Cf. A005249, A067689.

Cf. A001700. [Alexander Adamchuk, Nov 14 2009]

A060739 := n->(-1)^(n*(n-1)/2) * mul( (n+k-1)!/((k!)^2 * (n-1-k)!), k=0..n-1);

a[n_] := (-1)^(n (n - 1)/2)*Product[ Multinomial[k, k, n - 1 - k], {k, 0, n - 1}]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Dec 08 2011, after first formula *)

for(n=1,15,print1(1/matdet(matrix(n,n,i,j,i/(j+i))),",")) \\ See Cloitre's comment

{ for (n=0, 43, if (n<2, a=1, a=(-1)^(n\2)/matdet(matrix(n-1, n-1, i, j, i/(j+i)))); write("b060739.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 10 2009

def A060739(n): return (-1)^(n//2)*A163085(2*(n-1))/factorial(n-1) if n > 0 else 1
[A060739(i) for i in (0..11)] # Peter Luschny, Sep 18 2012

If Multinomial[a, b, c] denotes the multinomial coefficient (a+b+c)! / (a! * b! * c!) (which is an integer) then : a(n) = (-1)^(n(n-1)/2) * Product k=0, ..., n-1 Multinomial[k, k, n-1-k] = (-1)^(n(n-1)/2) * product k=0, ..., n-1 (n+k-1)!/((k!)^2 * (n-1-k)!)
a(n) = (-1)^(n(n-1)/2) * Product[ Binomial[2k,k]^2/2, {k,1,n-1} ]. [Alexander Adamchuk, Nov 14 2009]
|a(n)| = A163085(2*(n-1))/(n-1)! for n > 0. - Peter Luschny, Sep 18 2012
|a(n)| ~ A^3 * 2^(2*n^2 - 3*n + 5/12) * exp(n - 1/4) / (Pi^n * n^(n - 1/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 19 2020

A069651 For n >= 1, let M_n be the n X n matrix with M_n(i,j) = i^2/(i+j); then a(n) = 1/det(M_n). Also, a(0) = 1 by convention.

Original entry on oeis.org

1, 2, 18, 1200, 735000, 4667544000, 332086420512000, 279394363051195392000, 2892376010829659126572800000, 379850021025259936655866602240000000, 648304836222110631242066578424390188032000000

Offset: 0

Benoit Cloitre, Apr 21 2002

Also, determinant of the inverse of the (n+1)-st Hilbert matrix, divided by (2n+1)!. - Robert G. Wilson v, Feb 02 2004

Also, inverse of determinant of the matrix M_n(i,j) = i*j/(i+j). - Harry Richman, Aug 19 2019

Cf. A000142, A005249, A163085.

Table[1/((2n - 1)!Det[Table[1/(i + j - 1), {i, n}, {j, n}]]), {n, 10}] (* Robert G. Wilson v, Feb 02 2004 *)
Table[(n + 1)!/(2*n + 1)!*Product[Binomial[i, Floor[i/2]], {i, 1, 2*n + 1}], {n, 0, 10}] (* Stefan Steinerberger, Feb 26 2008 *)

for(n=1,15,print1(1/matdet(matrix(n,n,i,j,i^2/(j+i))),","))

def A069651(n): return A163085(2*n+1)/factorial(2*n+1)
[A069651(n) for n in (0..10)] # Peter Luschny, Sep 18 2012

a(n) = A005249(n)/A000142(n). - Robert G. Wilson v, Feb 02 2004
a(n) = (n+1)!/(2*n+1)! * Product[Binomial(i,Floor(i/2)), {i,1,2*n+1}]. - Stefan Steinerberger, Feb 26 2008
a(n) = A163085(2*n+1)/(2*n+1)! = A163085(2*n)/factorial(n)^2. - Peter Luschny, Sep 18 2012

Edited by N. J. A. Sloane, Feb 25 2008

A069945 Let M_k be the k X k matrix M_k(i,j)=1/binomial(i+n,j); then a(n)=1/det(M_(n+1)).

Original entry on oeis.org

1, -6, -360, 252000, 2222640000, -258768639360000, -410299414270986240000, 9061429740221589431500800000, 2835046804394206618956825845760000000, -12733381268715468286016211650968992153600000000

Offset: 1

Benoit Cloitre, Apr 27 2002

If k>n+1 det(M_k)=0

A005249, A056040.

a[n_] := (-1)^Quotient[n, 2]/(Det[HilbertMatrix[n]] n!); Array[a, 10] (* Jean-François Alcover, Jul 06 2019 *)

for(n=0,10,print1(1/matdet(matrix(n+1,n+1,i,j,1/binomial(i+n,j))),","))

def A069945(n): return (-1)^(n//2)*mul(binomial(i,i//2) for i in (1..2*n-1))
[A069945(i) for i in (1..11)] # Peter Luschny, Sep 18 2012

|a(n)| = det(M^(-1)), where M is an n X n matrix with M[i, j]=i/(i+j-1) (or j/(i+j-1)). |a(n)| = 1/det(HilbertMatrix(n))/n! = A005249(n)/n!. - Vladeta Jovovic, Jul 26 2003
|a(n)| = Product_{i=1..2n-1} binomial(i,floor(i/2)). - Peter Luschny, Sep 18 2012
|a(n)| = (Product_{i=1..2n-1} A056040(i))/n! = A163085(2*n-1)/n!. - Peter Luschny, Sep 18 2012

A120307 Inverse determinant of n X n matrix M[i,j] = i*j/(i+j-1).

Original entry on oeis.org

1, 3, 60, 10500, 18522000, 359400888000, 81408613942656000, 224737840779305293440000, 7812628980363223707442752000000, 3508978524227146242839564498172672000000

Offset: 1

Alexander Adamchuk, Jul 15 2006

nonn

Cf. A000108, A000891, A001700.

Table[ 1/Det[ Table[ i*j/(i+j-1), {i, n}, {j, n}]], {n,1,12}]

def A120307(n): return A163085(2*n)/factorial(2*n)
[A120307(n) for n in (1..10)] # Peter Luschny, Sep 18 2012

a(n) = 1/Det[ Table[ i*j/(i+j-1), {i, n}, {j, n}]]. a(n+1)/a(n) = A000891[n] = (2n)!(2n+1)! / (n! (n+1)!)^2 = (2n+1)*CatalanNumber[n]^2 = (2n+1)*A000108[n]^2 = C(2n+1,n+1)*CatalanNumber[n] = A001700[n]*A000108[n].

a(n) = A163085(2*n)/(2*n)!. - Peter Luschny, Sep 18 2012

cp's OEIS Frontend

A163086 Product of first n terms of A163085.

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A216628 a(n) = A163085(n)/n!.

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A216629 a(n) = A163085(n)/(floor(n/2)!)^2.

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A005249 Determinant of inverse Hilbert matrix.

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

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A060739 a(n) = (-1)^(n(n-1)/2) * Product_{k=0,...,n-1} (n+k-1)!/((k!)^2 * (n-1-k)!).

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A069651 For n >= 1, let M_n be the n X n matrix with M_n(i,j) = i^2/(i+j); then a(n) = 1/det(M_n). Also, a(0) = 1 by convention.

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