A005249 -id:A005249 - cp's OEIS frontend

A249185 Decimal expansion of a constant appearing in the Hankel determinant asymptotics.

6, 4, 5, 0, 0, 2, 4, 4, 8, 5, 0, 9, 5, 7, 7, 0, 8, 4, 6, 5, 8, 9, 6, 1, 0, 0, 7, 7, 2, 1, 7, 8, 7, 6, 5, 5, 3, 4, 7, 6, 1, 4, 4, 9, 4, 0, 5, 7, 3, 3, 9, 7, 2, 1, 5, 5, 2, 1, 4, 4, 5, 8, 8, 5, 8, 0, 2, 7, 6, 0, 7, 8, 7, 4, 1, 2, 4, 6, 8, 4, 6, 5, 7, 3, 9, 7, 1, 0, 5, 4, 9, 7, 1, 9, 7, 4, 0, 9, 9, 1, 4, 6

Offset: 0

Jean-François Alcover, Oct 23 2014

			0.645002448509577084658961007721787655347614494...

Steven Finch, Hankel and Toeplitz Determinants, March 17, 2014. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 416.
Eric Weisstein's MathWorld, Hilbert matrix
Wikipedia, Hilbert matrix

Cf. A005249, A074962.

evalf(limit(2^(1/12) * n^(3*n^2/2 + 3*n/2 + 1/4) * exp(1/4-3*n^2/4) / product(k^(3*k), k=1..n), n=infinity),120); # Vaclav Kotesovec, Oct 23 2014

h = 2^(1/12)*E^(1/4)*Glaisher^-3; RealDigits[h, 10, 102] // First

Det(H_n) ~ h*4^(-n^2)*(2*Pi)^n*n^(-1/4), where h = 2^(1/12)*e^(1/4)*A^(-3), A denoting the Glaisher-Kinkelin constant.

A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108).

Original entry on oeis.org

1, -2, -1400, -679140000, -122489812645200000, -6931927717187904217987200000, -114287375178291587421201860354580633600000, -527655997339226839875614785993553970321322576128000000000, -666218073328701414704702576237379472614149140939534461737723520000000000000

Offset: 1

Tom Richardson, Dec 03 2017

sign

It is conjectured that a(n) is an integer for all n.

The contributor suggests the name "Catbert matrix" for the matrix A_n, based on its similarity to the Hilbert matrix and its relation to the Catalan numbers.

Tom Richardson, Table of n, a(n) for n = 1..29
Tom Richardson, Table of n, a(n) for n = 1..100
Thomas M. Richardson, Catalan Numbers and Jacobi Polynomials, arXiv:2005.08939 [math.CO], 2020.

Cf. A000108, A005249, A062381.

a[n_] := 1/Det@ Table[ 1/CatalanNumber[i + j -2], {i, n}, {j, n}]; Array[a, 9] (* Robert G. Wilson v, Jan 05 2018 *)
Table[Product[4^(2*k + 1) * (4*k - 1)/6 * Binomial[2*k - 3/2, k] * Binomial[2*k - 3/2, k + 1], {k, 0, n - 1}], {n, 1, 10}] (* Vaclav Kotesovec, May 19 2020 *)

a(n) = 1/matdet(matrix(n,n,i,j,(i+j-1)/binomial(2*i+2*j-4,i+j-2)))

a(n) ~ -c * 16^(n*(n-1)) / (3^n * Pi^n * n^(27/8)), where c = 3*A^(3/2) / (2^(7/6) * exp(1/8) * sqrt(Pi)) = 0.9662886794923866798595701447717791386557874..., where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 19 2020

A348419 Triangular table read by rows: T(n,k) is the k-th entry of the main diagonal of the inverse Hilbert matrix of order n.

Original entry on oeis.org

1, 4, 12, 9, 192, 180, 16, 1200, 6480, 2800, 25, 4800, 79380, 179200, 44100, 36, 14700, 564480, 3628800, 4410000, 698544, 49, 37632, 2857680, 40320000, 133402500, 100590336, 11099088, 64, 84672, 11430720, 304920000, 2134440000, 4249941696, 2175421248, 176679360

Offset: 1

Jianing Song, Oct 18 2021

			The inverse Hilbert matrix of order 4 is given by
  [  16  -120   240  -140]
  [-120  1200 -2700  1680]
  [ 240 -2700  6480 -4200]
  [-140  1680 -4200  2800].
Hence the 4th row is 16, 1200, 6480, 2800.
The first 8 rows of the table are:
  1,
  4, 12,
  9, 192, 180,
  16, 1200, 6480, 2800,
  25, 4800, 79380, 179200, 44100,
  36, 14700, 564480, 3628800, 4410000, 698544,
  49, 37632, 2857680, 40320000, 133402500, 100590336, 11099088,
  64, 84672, 11430720, 304920000, 2134440000, 4249941696, 2175421248, 176679360,
  ...

Jianing Song, Table of n, a(n) for n = 1..5050 (first 100 rows)
Eric Weisstein's World of Mathematics, Hilbert Matrix

Cf. A189766 (row sums), A189765, A005249.
A210356 gives the maximum value of each row and A210357 gives the positions of the maximum values.
Main diagonal gives A000515(n-1).

T:= n-> (M-> seq(M[i, i], i=1..n))(1/LinearAlgebra[HilbertMatrix](n)):
seq(T(n), n=1..8);  # Alois P. Heinz, Jun 19 2022

T[n_, k_] := Inverse[HilbertMatrix[n]][[k, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 18 2021 *)

PARI
```
T(n,k) = (1/mathilbert(n))[k,k]
```

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.

A111237 LCM of the absolute values of the inverse Hilbert matrix.

Original entry on oeis.org

1, 12, 2880, 226800, 101606400, 6985440000, 35961045120000, 1431699108840000, 692306057963520000, 181128033944995737600, 344143264495491901440000, 6651046563131624734080000, 22028266217091941119272960000

Offset: 1

Roger Hui, Oct 28 2005

nonn

Cf. A005249 (determinant), A111194 (permanent), A092326 (permanent/determinant).

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

A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).

Original entry on oeis.org

2, 5, 1, 17, 13, 37, 1, 1, 41, 101, 61, 29, 1, 197, 113, 257, 1, 1, 181, 401, 1, 97, 53, 577, 313, 677, 73, 157, 421, 1, 1, 1, 109, 89, 613, 1297, 137, 1, 761, 1601

Offset: 1

Alexander Adamchuk, Jul 10 2006

Some a(n) are equal to 1 (for n=3,7,8,13,17,18,21,30,31,32,38..=A002312 Arc-cotangent reducible numbers or non-Stormer numbers). All other a(n) (for n=1,2,4,5,6,9,10,11,14,15,16,19,20,22,23..=A005528 Stormer numbers or arc-cotangent irreducible numbers, largest prime factor of n^2 + 1 is >= 2n.) belong to A005529 - Primitive prime factors of the sequence k^2 + 1 (A002522) in the order that they are found. Matrix M[i,j] = (i+j)/(i+j-1) = 1 + 1/(i+j-1) is a sum of n X n unit matrix and n X n Hilbert Matrix. Denominator of determinant of matrix M[i,j] equals determinant of inverse Hilbert matrix A005249.

Cf. A005249, A002312, A005528, A005529, A002522.

Numerator[Table[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]],{n,1,40}]]

a(n) = numerator[Det[Table[(i+j)/(i+j-1),{i,1,n},{j,1,n}]]].

cp's OEIS Frontend

A249185 Decimal expansion of a constant appearing in the Hankel determinant asymptotics.

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A296056 Determinant of the inverse of the matrix A_n, where A_n is the n X n matrix defined by A_n[i,j] = 1/C(i+j-2) for 1 <= i,j <= n, and C(k) is the k-th Catalan number (A000108).

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A348419 Triangular table read by rows: T(n,k) is the k-th entry of the main diagonal of the inverse Hilbert matrix of order n.

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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).

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A111237 LCM of the absolute values of the inverse Hilbert matrix.

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J

A120294 Numerator of determinant of n X n matrix with elements M[j,j] = (i+j)/(i+j-1).

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Formula