cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A322908 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

Original entry on oeis.org

1, -5, 38, -386, 4928, -75927, 1371808, -28452356, 666445568, -17402398505, 501297595904, -15792876550662, 540190822408192, -19937252888438459, 789770307546718208, -33422580292067020808, 1504926927960887066624, -71839548181524098808909, 3624029163661165580910592
Offset: 1

Views

Author

Stefano Spezia, Dec 30 2018

Keywords

Comments

The matrix M(n) differs from that of A318173 in using successive positive integers in place of successive prime numbers.
The trace of the matrix M(n) is A000027(n).
The sum of the first row of the matrix M(n) is A000217(n).
The sum of the first column of the matrix M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 11 2019]
For n > 1, the sum of the superdiagonal of the matrix M(n) is A005843(n).

Examples

			For n = 1 the matrix M(1) is
   1
with determinant Det(M(1)) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with Det(M(2)) = -5.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with Det(M(3)) = 38.

Links

Crossrefs

Cf. A000027, A000217, A005448, A005843, A318173.
Cf. A322909 (permanent of matrix M(n)).

Programs

  • Maple
    a:= proc(n) uses LinearAlgebra;
Determinant(ToeplitzMatrix([seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))
end proc:
map(a, [$1..20]);
  • Mathematica
    b[n_]:=n; a[n_]:=Det[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]; Array[a, 20]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matdet(tm(n)); \\ Michel Marcus, Nov 11 2020

Formula

a(n) ~ -(-1)^n * (3*exp(1) - exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 05 2019

A323254 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

Original entry on oeis.org

1, 7, 58, 614, 8032, 125757, 2298208, 48075148, 1133554432, 29756555315, 860884417024, 27218972906226, 933850899349504, 34556209025624041, 1371957513591119872, 58174957356247084568, 2624017129323317493760, 125454378698728779884895, 6337442836338834419089408
Offset: 1

Views

Author

Stefano Spezia, Jan 09 2019

Keywords

Comments

The trace of the matrix M(n) is A000384(n). [Corrected by Stefano Spezia, Dec 08 2019]
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

Examples

			For n = 1 the matrix M(1) is
   1
with determinant Det(M(1)) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with Det(M(2)) = 7.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with Det(M(3)) = 58.

Links

Crossrefs

Cf. A000290, A000326, A000384, A001105, A001792.
Cf. A034856, A204235, A318173, A322908, A322909.
Cf. A323255 (permanent of matrix M(n)).

Programs

  • Mathematica
    b[i_]:=i; a[n_]:=Det[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n}]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]; Array[a,20]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matdet(tm(n)); \\ Stefano Spezia, Dec 11 2019

Formula

a(n) ~ (5*exp(1) + exp(-1)) * n^n / 4. - Vaclav Kotesovec, Jan 10 2019

A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

Original entry on oeis.org

1, 1, 11, 248, 9968, 638772, 60061657, 7798036000, 1336715859150, 292406145227392, 79483340339739367, 26280500564448081664, 10386012861097225139356, 4834639222955142417477888, 2618110215141486526589786501, 1631888040186649673361825042432, 1159983453675106278249250918734938
Offset: 0

Views

Author

Stefano Spezia, Jan 09 2019

Keywords

Comments

The trace of the matrix M(n) is A000384(n).
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

Examples

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with permanent a(3) = 248.

Links

Crossrefs

Cf. A000290, A000326, A000384, A001105, A001792.
Cf. A034856, A204235, A318173, A322908, A322909.
Cf. A323254 (determinant of matrix M(n)).

Programs

  • Mathematica
    b[i_]:=i; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n }]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]]; Array[a, 16, 0]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 11 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 08 2019
Showing 1-3 of 3 results.

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