A005249 -id:A005249 - cp's OEIS frontend

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)).

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

A101029 Denominator of partial sums of a certain series.

Original entry on oeis.org

1, 10, 70, 420, 4620, 60060, 60060, 408408, 7759752, 38798760, 892371480, 4461857400, 13385572200, 55454513400, 1719089915400, 3438179830800, 24067258815600, 890488576177200, 890488576177200, 36510031623265200, 1569931359800403600, 1569931359800403600, 73786773910618969200

Offset: 1

Wolfdieter Lang, Dec 17 2004

The numerators are given in A101028.

One third of the denominator of the finite differences of the series of sums of all matrix elements of n X n Hilbert matrix M(i,j)=1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 11 2006

			n=2: HilbertMatrix[n,n]
   1  1/2
  1/2 1/3
so a(1) = (1/3)*denominator((1 + 1/2 + 1/2 + 1/3) - 1) = (1/3)*denominator(4/3) = 1.
The n X n Hilbert 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 ...

Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A098118, A086881, A005249, A001008, A002805.

Cf. A101028 (numerators).

Denominator[Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n,2, 27}]-Table[Sum[1/(i + j - 1), {i, n}, {j, n}], {n, 26}]]/3 (* Alexander Adamchuk, Apr 11 2006 *)

a(n) = denominator(3*sum(k=1, n, 1/((2*k-1)*k*(2*k+1)))); \\ Michel Marcus, Feb 28 2022

a(n) = denominator(s(n)) with s(n) = 3*Sum_{k=1..n} 1/((2*k-1)*k*(2*k+1)). See A101028 for more information.

a(n) = (1/3)*denominator((Sum_{i=1..n+1} Sum_{j=1..n+1} 1/(i+j-1)) - (Sum_{i=1..n} Sum_{j=1..n} 1/(i+j-1))). a(n) = (1/3)*denominator(H(2*n+1) + H(2*n) - 2*H(n)), where H(n) = Sum_{k=1..n} 1/k is a harmonic number, H(n) = A001008/A002805. - Alexander Adamchuk, Apr 11 2006

More terms from Michel Marcus, Feb 28 2022

A278840 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000041(i+j).

Original entry on oeis.org

1, 2, 19, 642, 58884, 13569779, 6931351962, 7532494931779, 16299546505518855, 67814300022651169814, 520884812091898994319805, 7206655416715261673779120809, 174009567319884878178189603283634, 7196671016523025599652036668556922867

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Cf. A000041, A005249, A204249, A278838, A278841.

Flatten[{1, Table[Permanent[Table[PartitionsP[i+j], {i, 1, n}, {j, 1, n}]], {n, 1, 14}]}]

A278841 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000009(i+j).

Original entry on oeis.org

1, 1, 6, 65, 1737, 91359, 8755730, 1465091787, 420070484342, 194857695312573, 142349343815684947, 161388097061567486595, 276760372468557882285737, 707850058213409589011565269, 2654427644322345709705054800083

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Cf. A000009, A005249, A204249, A278839, A278840.

Flatten[{1, Table[Permanent[Table[PartitionsQ[i+j], {i, 1, n}, {j, 1, n}]], {n, 1, 14}]}]

A061913 Numerators of the determinant of matrix (M(n) - H(n)), where H(n) is the n-th Hilbert matrix and M(n) is an n X n matrix with i,j-th entry i+j-1.

Original entry on oeis.org

1, 0, -9, 32, -337, 319, -347, 1609, -21569, 21911, -82601, 12211, -247249, 50003, -624457, 16297, -42209, 91127, -91159, 488749, -1773467, 1782853, -9429289, 6156767, -548213, 376169, -25705577, 732761, -84793, 343823, -60683401, 36941321, -89392129, 53762297, -1478431

Offset: 0

Asher Auel, May 20 2001

The n-th Hilbert matrix is an n X n matrix with i,j-th entry 1/(i+j-1). Thus (M(n) - H(n)) has i,j-th entry (i+j-1) - 1/(i+j-1).

Cf. A005249, A380297 (denominators).

a[n_]:=Numerator[Det[Table[(i+j-1) - 1/(i+j-1),{i,n},{j,n}]]]; Join[{1},Array[a,34]] (* Stefano Spezia, Jan 19 2025 *)

a(n) = numerator(matdet(matrix(n, n, i, j, (i+j-1) - 1/(i+j-1)))); \\ Michel Marcus, Jan 19 2025

a(0) = 1 prepended by Stefano Spezia, Jan 19 2025

A061914 Let H_n = n-th Hilbert matrix; sequence gives 1 / ( det(H_n) * denominator(permanent(H_n)) ).

Original entry on oeis.org

1, 1, 1, 27, 567, 1, 1, 1, 7, 9, 5103, 1275989841, 992436543, 48629390607, 169706648853, 40257567, 63, 1, 7, 31, 1, 3969, 25865973, 117649, 117649, 16807, 49, 9, 81, 117369, 59049, 33480783, 930196594089, 4238886345135097131, 169560200598623521407

Offset: 1

Asher Auel, May 20 2001

nonn

Eric Weisstein's World of Mathematics, Permanent.

Cf. A005249.

with(linalg): seq(1/(denom(permanent(hilbert(n)))*det(hilbert(n))), n=1..16);

Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m.v), Times @@ v]]; f[n_] := Block[{i = Table[1/(i + j - 1), {i, n}, {j, n}]}, 1/(Det[i]Denominator[Permanent[i]])]; Table[ f[n], {n, 1, 18}] (* Robert G. Wilson v, Feb 06 2004 *)

permRWN(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); n1=n-1; sg=1; m=1; nc=0; in=vector(n); x=in; for(i=1,n,x[i]=a[i,n]-sum(j=1,n,a[i,j])/2); p=prod(i=1,n,x[i]); while(m,sg=-sg; j=1; if((nc%2)!=0,j++; while(in[j-1]==0,j++)); in[j]=1-in[j]; nc+=2*in[j]-1; m=nc!=in[n1]; z=2*in[j]-1; for(i=1,n,x[i]+=z*a[i,j]); p+=sg*prod(i=1,n,x[i])); return(2*(2*(n%2)-1)*p)
for(n=1,23,a=mathilbert(n); print1(1/(matdet(a)*denominator(permRWN(a)))", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007

for(n=1, 25, a=mathilbert(n); print1(1 / (matdet(a) * denominator(matpermanent(a)))", ")) \\ Vaclav Kotesovec, Aug 13 2021

a(n) = 1/(denominator(permanent(hilbert(n)))*det(hilbert(n))), where hilbert(n) denotes the n-th Hilbert matrix.

a(18)-a(20) from Robert G. Wilson v, Feb 09 2004
a(21) from Eric W. Weisstein, Feb 19 2004
a(22) and a(23) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 10 2007
a(24)-a(34) from Vaclav Kotesovec, Aug 14 2021
a(35) from Vaclav Kotesovec, Aug 16 2021

A076823 Array of coefficients of 1/det(M_n)*P(M_n) where P(M_n) is the characteristic polynomial of the n-th n X n Hilbert matrix M_n(i,j)=1/(i+j-1).

Original entry on oeis.org

-1, 1, 1, -16, 12, -1, 381, -3312, 2160, 1, -10496, 1603680, -10137600, 6048000, -1, 307505, -1022881200, 92708406000, -476703360000, 266716800000, 1, -9316560, 750409713900, -1242627237734400, 78981336366912000, -349935855575040000, 186313420339200000, -1

Offset: 1

Benoit Cloitre, Nov 27 2002

Montgomery made a conjecture related to the largest eigenvalue of the Hilbert matrix (cf. Matthews link)

			Triangle begins:
  -1, 1;
  1, -16, 12;
  -1, 381, -3312, -2160;
  ...

Robert Israel, Table of n, a(n) for n = 1..902
Keith Matthews, Hilbert inequality.

Cf. A005249.

f:= proc(n) uses LinearAlgebra; local P,M;
  M:= HilbertMatrix(n);
  P:= CharacteristicPolynomial(M,t)/Determinant(M);
  seq(coeff(P,t,i),i=0..n)
end proc:
seq(f(n),n=1..10); # Robert Israel, May 07 2018

row[n_] := Module[{P, M, x}, M = HilbertMatrix[n]; P = CharacteristicPolynomial[M, x]/Det[M]; (-1)^n CoefficientList[P, x]];
Array[row, 10] // Flatten (* Jean-François Alcover, Jun 22 2020 *)

vector(n+1,i,(polcoeff(charpoly(mathilbert(n))/matdet(mathilbert(n)),i-1))) \\ for the "n-th row"

T(n,0)=(-1)^n, T(n,n) = A005249(n). - Robert Israel, May 07 2018

A111194 Permanent of the inverse Hilbert matrix.

Original entry on oeis.org

1, 1, 84, 1397520, 5314794912000, 4855173934730716800000, 1090093558153665322315192780800000, 60907190511553979457004412118419080463155200000

Offset: 0

Roger Hui, Oct 22 2005

nonn

Cf. A005249 = determinant of inverse Hilbert matrix; and A092326 = (permanent/determinant) of inverse Hilbert matrix.

NB. www.jsoftware.com
H =: % @: >: @: (+/~) @: i. @ x:
perm=: +/ .*
perm@%.@H n

Permanent[m_List] := With[{v = Array[x, Length[m]]}, Coefficient[Times @@ (m . v), Times @@ v]]; f[n_] := Block[{i = Inverse[Table[1/(i + j - 1), {i, n}, {j, n}]]}, Permanent[i]]; Table[ f[n], {n, 7}] (* Robert G. Wilson v, Oct 24 2005 *)

A135451 Triangular function from the characteristic polynomials of the inverse Hilbert matrices.

Original entry on oeis.org

1, 1, -1, 12, -16, 1, 2160, -3312, 381, -1, 6048000, -10137600, 1603680, -10496, 1, 266716800000, -476703360000, 92708406000, -1022881200, 307505, -1, 186313420339200000, -349935855575040000, 78981336366912000, -1242627237734400, 750409713900, -9316560, 1

Offset: 0

Roger L. Bagula, Dec 14 2007

Triangle read by rows: for 0 <= k <= n, T(n,k) is the coefficient of lambda^k in det(H^(-1) - lambda I) where H is the n x n Hilbert matrix.

Row sums are: 1, 0, -3, -772, -2496415, -118300727696, -85882975706265059, -972835586209103886374316, -173520203650301344466515679407359, -489847775570499454780372858733881836257416, -21954569246037949585920541114453120558720536422853379

			{1},
{1, -1},
{12, -16, 1},
{2160, -3312, 381, -1},
{6048000, -10137600, 1603680, -10496, 1},
{266716800000, -476703360000, 92708406000, -1022881200, 307505, -1},
{186313420339200000, -349935855575040000, 78981336366912000, -1242627237734400, 750409713900, -9316560, 1}

Robert Israel, Table of n, a(n) for n = 0..902 (rows 0 to 41, flattened)
Eric Weisstein's World of Mathematics, Hilbert matrix

Cf. A005249.

f:= proc(n) uses LinearAlgebra;
local lambda, P,j;
P:= CharacteristicPolynomial(HilbertMatrix(n),lambda)/Determinant(HilbertMatrix(n));
seq(coeff(P,lambda,n-j),j=0..n);
end proc:
seq(f(n),n=0..10); # Robert Israel, Oct 05 2016

<< LinearAlgebra`MatrixManipulation`; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x], {n, 1, 10}]]; Flatten[a]

t(n,m)=CoefficientList[CharacteristicPolynomial[Inverse[HilbertMatrix[n]], x], x]

Edited by Robert Israel, Oct 05 2016

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

cp's OEIS Frontend

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)).

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A101029 Denominator of partial sums of a certain series.

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A278840 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000041(i+j).

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A278841 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000009(i+j).

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A061913 Numerators of the determinant of matrix (M(n) - H(n)), where H(n) is the n-th Hilbert matrix and M(n) is an n X n matrix with i,j-th entry i+j-1.

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A061914 Let H_n = n-th Hilbert matrix; sequence gives 1 / ( det(H_n) * denominator(permanent(H_n)) ).

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A076823 Array of coefficients of 1/det(M_n)*P(M_n) where P(M_n) is the characteristic polynomial of the n-th n X n Hilbert matrix M_n(i,j)=1/(i+j-1).

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A111194 Permanent of the inverse Hilbert matrix.

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