A189766 -id:A189766 - cp's OEIS frontend

A178790 The arithmetic mean of (2k+1)A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

1, 8, 127, 2624, 61501, 1552760, 41186755, 1131614720, 31923047665, 919243356008, 26908963456783, 798379043762624, 23954974906866901, 725620080605773592, 22159617936375571627, 681528994326392115200, 21090805673899997148025, 656256696917886135153800

Offset: 1

Zhi-Wei Sun, Jun 14 2010

nonn

Conjecture: the number a(n) = n^(-1)*Sum_{k=0..n-1}(2*k+1)*A_k is always an integer.
We can prove that for any prime p>3 we have a(p)=p (mod p^4).
Conjecture: If p=5,7 (mod 8) is a prime then sum_{k=0}^{p-1}A_k=0 (mod p^2); if p=1,3 (mod 8) is a prime greater than 3 and p=x^2+2y^2 with x,y integers then sum_{k=0}^{p-1}A_k=4x^2-2p (mod p^2).
a(n) is always an integer. The detailed proof can be found in the latest version of arXiv:1006.2776 . - Zhi-Wei Sun, Jun 17 2010

			For n=3 we have a(3)=(A_0+3A_1+5A_2)/3=(1+3*5+5*73)/3=127.

G. C. Greubel, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, arXiv:1006.2776 [math.NT], 2011.

Cf. A005259, A173774, A189766.

List([1..30], n-> Sum([0..n], k -> Binomial(n-1,k)*Binomial(n+k,k) *Binomial(n+k, 2*k+1)* Binomial(2*k,k) )); # G. C. Greubel, Jan 24 2019

[(&+[Binomial(n-1,k)*Binomial(n+k,k)*Binomial(n+k, 2*k+1)* Binomial(2*k,k): k in [0..n-1]]): n in [1..30]]; // G. C. Greubel, Jan 24 2019

G := (-1/2)*(3*x-3+(x^2-34*x+1)^(1/2))*(x+1)^(-2)*hypergeom([1/3,2/3],[1],(-1/2)*(x^2-7*x+1)*(x+1)^(-3)*(x^2-34*x+1)^(1/2)+(1/2)*(x^3+30*x^2-24*x+1)*(x+1)^(-3))^2;
ogf := 2*x*G/(1-x)-Int((x+1)*G/(x-1)^2,x);
series(ogf,x=0,25); # Mark van Hoeij, May 07 2013

Apery[n_]:= Sum[Binomial[n+k,k]^2 Binomial[n,k]^2,{k,0,n}]; AA[n_]:= Sum[(2k+1)*Apery[k],{k,0,n-1}]/n; Table[AA[n],{n,1,25}]
Table[Sum[(Binomial[n-1,k]*Binomial[n+k,k]*Binomial[n+k,2*k+1]* Binomial[2*k,k]), {k,0,n-1}], {n,1,30}] (* G. C. Greubel, Jan 24 2019 *)

{a(n) = sum(k=0,n-1, binomial(n-1,k)*binomial(n+k,k)*binomial(n+k, 2*k+1)*binomial(2*k,k))}; \\ G. C. Greubel, Jan 24 2019

[sum(binomial(n-1,k)*binomial(n+k,k)*binomial(n+k, 2*k+1)* binomial(2*k,k) for k in (0..n-1)) for n in (1..30)] # G. C. Greubel, Jan 24 2019

Recursion : (n+2)^3 *(n+3) *(2n+1) *a(n+3) = (n+2) *(2n+1) *(35*n^3+193*n^2+345*n+203) *a(n+2) -(n+1) *(2n+5) *(35*n^3+122*n^2+132*n+40) * a(n+1) +n *(n+1)^3 *(2n+5) *a(n).
a(n) = Sum(k=0..n-1, (binomial(n-1,k)* binomial(n+k,k)* binomial(n+k,2*k+1)*binomial(2*k,k)) ). - Zhi-Wei Sun, Jun 17 2010
a(n) = A189766(n) / n = trace( HilbertMatrix(n)^(-1) )/n. - Richard Penner, Jun 04 2011
a(n) = (1/n)*Sum_{k=0..n-1} (2*k+1)*binomial(n+k,2*k+1)^2*binomial(2*k, k)^2. - Richard Penner, Jun 04 2011
G.f.: 2*x*G/(1-x)-Int((x+1)*G/(x-1)^2,x) where G is the generating function of A005259. - Mark van Hoeij, May 07 2013
a(n) ~ 2^(1/4) * (1 + sqrt(2))^(4*n) / (16*(Pi*n)^(3/2)). - Vaclav Kotesovec, Jan 24 2019

A189765 Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.

Original entry on oeis.org

1, 4, -6, 12, 9, -36, 192, 30, -180, 180, 16, -120, 1200, 240, -2700, 6480, -140, 1680, -4200, 2800, 25, -300, 4800, 1050, -18900, 79380, -1400, 26880, -117600, 179200, 630, -12600, 56700, -88200, 44100, 36, -630, 14700, 3360, -88200, 564480, -7560, 211680

Offset: 1

T. D. Noe, May 02 2011

The n-th order Hilbert matrix has elements h(i,j) = 1/(i+j-1) for 1 <= i,j <=n. Only the lower triangular matrix is shown because the Hilbert matrix and its inverse are symmetric. The n-th row begins with n^2 and ends with A000515(n+1).
The sums of select rows of the inverse matrix are sequences A002457, A002736, A002738, A007531, and A054559.
The largest magnitude in the matrix is A210356(n). - T. D. Noe, Mar 28 2012
The sum of the elements of the n-th matrix is n^2. - T. D. Noe, Apr 02 2012

			Row 3 is 9, -36, 192, 30, -180, 180 which corresponds to the inverse
  9  -36   30
-36  192 -180
30 -180  180

T. D. Noe, Rows n = 1..25, flattened
Eric W. Weisstein, MathWorld: Hilbert Matrix

Cf. A002457, A002736, A002738, A005249 (determinant), A007531, A054559, A189766 (trace).

lowerTri[m_List] := Module[{n = Length[m]}, Flatten[Table[Take[m[[i]], i], {i, n}]]]; Flatten[Table[lowerTri[Inverse[HilbertMatrix[n]]], {n, 6}]]

a(n,i,j) = (-1)^(i+j) (i+j-1) binomial(n+i-1, n-j) binomial(n+j-1, n-i) binomial(i+j-2, i-1)^2 is the (i,j) element of the inverse of the n-th Hilbert matrix.

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]
```

cp's OEIS Frontend

A178790 The arithmetic mean of (2k+1)A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

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A189765 Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.

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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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cp's OEIS Frontend

A178790 The arithmetic mean of (2*k+1)*A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.

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Author

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Examples

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Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A189765 Triangle in which row n has the n(n+1)/2 elements of the lower triangular part of the inverse of the n-th order Hilbert matrix.

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A178790 The arithmetic mean of (2k+1)A_k (k=0,...,n-1), where A_0,A_1,... are Apery numbers given by A005259.