A067689 -id:A067689 - cp's OEIS frontend

A005249 Determinant of inverse Hilbert matrix.

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

A003046 Product of first n Catalan numbers.

Original entry on oeis.org

1, 1, 2, 10, 140, 5880, 776160, 332972640, 476150875200, 2315045555222400, 38883505145515430400, 2285805733484270091494400, 475475022233529990271933132800, 353230394017289429773019124357120000, 944693494975599542562153266945656012800000

Offset: 0

N. J. A. Sloane

The volume of a certain polytope (see Chan et al. reference). However, no combinatorial explanation for this is known.

H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..60
V. Baldoni and M. Vergne, Kostant Partitions Functions and Flow Polytopes, Transform. Groups. 13 (2008), 447-469.
C. S. Chan et al., On the volume of a certain polytope, Experimental Mathematics, 9 (2000), 91-99.
S. Corteel, J. S. Kim and K. Mészáros, Flow polytopes with Catalan volumes, C. R. Math., 355 (2017), 248-259.
Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.
K. Mészáros and A. H. Morales, Flow polytopes of signed graphs and the Kostant partition function, IMRN 3 (2015), 830-871.
J. W. Moon and M. Sobel, Enumerating a class of nested group testing procedures, J. Combin. Theory, B23 (1977), 184-188.
J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973
D. Zeilberger, Proof of a Conjecture of Chan, Robbins and Yuen, arXiv:math/9811108 [math.CO], 1998.
Index to divisibility sequences

Cf. A003047, A000108, A055746, A069640, A005249, A051575, A067689, A074962.

a003046 n = a003046_list !! n
a003046_list = scanl1 (*) a000108_list
-- Reinhard Zumkeller, Oct 01 2012

seq(mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13); # Zerinvary Lajos, Jul 02 2008

a[n_] := Product[ CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 05 2012 *)
FoldList[Times,1,CatalanNumber[Range[20]]] (* Harvey P. Dale, Apr 29 2013 *)
Table[(2^(n^2+n-1/24) Glaisher^(3/2) BarnesG[n+3/2])/(Exp[1/8] Pi^(n/2+1/4) BarnesG[n+3]), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 11 2015 *)

a(n) = prod(k=0, n, binomial(2*k,k)/(k+1)); \\ Michel Marcus, Sep 06 2021

a(n) = C(0)*C(1)*...*C(n), C() = A000108 = Catalan numbers.
a(n) = sqrt((2^n)*A069640(n)/(2*n+1)!/n!), n>0, where A069640(n) is an inverse determinant of the n X n Hilbert-like Matrix with elements M(i,j)=1/(i+j+1). - Alexander Adamchuk, May 17 2006
a(n) ~ A^(3/2) * 2^(n^2 + n - 19/24) * exp(3*n/2 - 1/8) / (n^(3*n/2 + 15/8) * Pi^(n/2+1)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014
a(n) = A^(3/2)*2^(n^2 + n - 1/24)*BarnesG(n+3/2) / (exp(1/8)*Pi^(n/2 + 1/4)*BarnesG(n+3)), where BarnesG( ) is the Barnes G-function and A is the Glaisher-Kinkelin constant (A074962). - Ilya Gutkovskiy, Mar 16 2017
For n > 0, a(n) = 2^((n+1)/2) * sqrt(BarnesG(2*n)) * Gamma(2*n) / (n^2 * (n+1) * BarnesG(n)^2 * Gamma(n)^(9/2)). - Vaclav Kotesovec, Nov 27 2024

a(15) added by Harvey P. Dale, Apr 29 2013
Typo in second formula corrected by Vaclav Kotesovec, Nov 13 2014
Links added by Alejandro H. Morales, Jan 26 2020

A163085 Product of first n swinging factorials (A056040).

Original entry on oeis.org

1, 1, 2, 12, 72, 2160, 43200, 6048000, 423360000, 266716800000, 67212633600000, 186313420339200000, 172153600393420800000, 2067909047925770649600000, 7097063852481244869427200000

Offset: 0

Peter Luschny, Jul 21 2009

nonn

With the definition of the Hankel transform as given by Luschny (see link) which uniquely determines the original sequence (provided that all determinants are not zero) this is also 1/ the Hankel determinant of 1/(n+1) (assuming (0,0)-based matrices).
a(2*n-1) is 1/determinant of the Hilbert matrix H(n) (A005249).
a(2*n) = A067689(n). - Peter Luschny, Sep 18 2012

Peter Luschny, SequenceTransformations

Cf. A056040, A163086, A055462, A000178.

a := proc(n) local i; mul(A056040(i),i=0..n) end;

a[0] = 1; a[n_] := a[n] = a[n-1]*n!/Floor[n/2]!^2; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 26 2013 *)

def A056040(n):
    swing = lambda n: factorial(n)/factorial(n//2)^2
    return mul(swing(i) for i in (0..n))
[A056040(i) for i in (0..14)] # Peter Luschny, Sep 18 2012

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

A069640 Let M_n be the n X n matrix with M_n(i,j)=1/(i+j+1); then a(n)=1/det(M_n).

Original entry on oeis.org

3, 240, 378000, 10668672000, 5175372787200000, 42202225467872870400000, 5708700736339601341845504000000, 12701009683686045652926579789004800000000, 462068939479146913162956288390362787269836800000000

Offset: 1

Benoit Cloitre, Apr 21 2002

Wolfram Research, 1991 Mathematica Conference, Elementary Tutorial Notes, Section 1, Introduction to Mathematica, Paul Abbott, page 19.

Robert Israel, Table of n, a(n) for n = 1..40

Cf. A000108, A003046, A005249, A067689, A074962.

seq(1/LinearAlgebra:-Determinant(LinearAlgebra:-HilbertMatrix(n,n,-1)),n=1..10); # Robert Israel, Sep 26 2018

Hilbert[n_Integer] := Table[1/(i + j + 1), {i, n}, {j, n}]; Table[ 1 / Det[ Hilbert[n]], {n, 1, 8}] (* Robert G. Wilson v, Mar 13 2004 *)
Table[(2*n+1)!!*(n!*Product[(2*k)!/k!/(k+1)!,{k,0,n}])^2,{n,1,11}] (* Alexander Adamchuk, May 17 2006 *)
Table[2^(2*n^2+2*n-1/12) * Glaisher^3 * BarnesG[n+3/2]^2 *(n!)^2 *(2*n+1)!!/(E^(1/4)*Pi^(n+1/2)*BarnesG[n+3]^2), {n, 1, 11}] (* Vaclav Kotesovec, Mar 09 2014 *)

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

a(n) = (2*n+1)!!*(n!*Product[(2*k)!/k!/(k+1)!,{k,0,n}])^2. a(n) = (2*n+1)!!*(n!*A003046(n))^2, where A003046(n)is the Product of first n Catalan numbers A000108(n). a(n) = (2*n+1)!*n!/(2^n)*A003046(n)^2. - Alexander Adamchuk, May 17 2006

a(n) ~ A^3 * 2^(2*n^2+3*n+11/12) / (exp(1/4) * n^(7/4) * Pi^(n+1)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

A174841 Determinant of the symmetric n X n matrix M_n where M_n(j,k) = n^abs(j-k).

Original entry on oeis.org

1, -3, 64, -3375, 331776, -52521875, 12230590464, -3938980639167, 1677721600000000, -913517247483640899, 619173642240000000000, -511324276025564512546607, 505488617542763051300683776

Offset: 1

Michel Lagneau, Mar 30 2010

sign

			a(4) = determinant(M_4) = -3375 where M_4 is the matrix
[ 1  4 16 64]
[ 4  1  4 16]
[16  4  1  4]
[64 16  4  1]

Jerry Glynn and Theodore Gray, The Beginner's Guide to Mathematica Version 4, Cambridge University Press, 2000, p. 76.

Vincenzo Librandi, Table of n, a(n) for n = 1..76

Cf. A005249, A067689.

[ Determinant( SymmetricMatrix( &cat[ [ n^Abs(j-k): k in [1..j] ]: j in [1..n] ] ) ): n in [1..13] ]; // Klaus Brockhaus, Apr 16 2010

for n from 1 to 20 do: x:=(1-n^2)^(n-1):print(x):od:

a(n) = (1-n^2)^(n-1).

Edited by Klaus Brockhaus, Apr 16 2010

A069704 Let M_2n be the 2n X 2n matrix M_(i,j)=C(2i,j)-C(2j,i) where C(k,l) denotes the binomial coefficients; then a(n)=det(M_2n).

Original entry on oeis.org

9, 2401, 7139584, 7429060864, 8768304271322176, 55287838983612748529926144, 28712457542131501655125523112656896, 597353290725130889841630014909751870078582784

Offset: 1

Benoit Cloitre, Apr 21 2002

det(M_{2n+1})=0

Cf. A005249, A060739, A067689.

A090914 Reciprocal of (n+1)! times determinant of n X n matrix whose (i,j)-th element is 1/(i+j).

Original entry on oeis.org

1, 1, 12, 1800, 3528000, 93350880000, 34157460395520000, 176018448722253096960000, 12957844528516872887046144000000, 13783997562964632581368087262085120000000

Offset: 0

Robert G. Wilson v, Feb 26 2004

nonn

A067689/(n+1)!

Table[ 1 / Det[ Table[ 1 / (i + j), {i, 1, n}, {j, 1, n} ]]/(n + 1)!, {n, 1, 10}]

a(n)=if(n<0,0,1/((n+1)!*matdet(matrix(n,n,i,j,1/(i+j)))))

A094592 a(n)=1/det(M(n)) where M(n) is the n X n matrix with element 1/binomial(n+i+j,n).

Original entry on oeis.org

1, 3, 900, 16464000, 19061082810000, 1438322180987553878016, 7234500281404807826660044800000, 2467969483765439710971533630106501120000000

Offset: 0

Benoit Cloitre, Jun 05 2004

nonn

Cf. A062381, A067689.

a(n)=1/matdet(matrix(n,n,i,j,1/binomial(n+i+j,n)))

cp's OEIS Frontend

A005249 Determinant of inverse Hilbert matrix.

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A003046 Product of first n Catalan numbers.

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A163085 Product of first n swinging factorials (A056040).

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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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A069640 Let M_n be the n X n matrix with M_n(i,j)=1/(i+j+1); then a(n)=1/det(M_n).

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A174841 Determinant of the symmetric n X n matrix M_n where M_n(j,k) = n^abs(j-k).

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