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 1-5 of 5 results.

A302909 Determinant of n X n matrix whose main diagonal consists of the first n 5-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 4, 44, 924, 31416, 1570800, 108385200, 9863053200, 1144114171200, 164752440652800, 28831677114240000, 6025820516876160000, 1482351847151535360000, 423952628285339112960000, 139480414705876568163840000, 52305155514703713061440000000
Offset: 1

Views

Author

Muniru A Asiru, Apr 15 2018

Keywords

Examples

			The 7 X 7 matrix (as below) has determinant 108385200.
  1  1  1  1  1  1  1
  1  5  1  1  1  1  1
  1  1 12  1  1  1  1
  1  1  1 22  1  1  1
  1  1  1  1 35  1  1
  1  1  1  1  1 51  1
  1  1  1  1  1  1 70

Links

Crossrefs

Cf. A000326, A115067.
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), this sequence (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

Programs

  • Maple
    d:=(i,j)->`if`(i<>j,1,i*(3*i-1)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..17);
  • Mathematica
    Table[FullSimplify[Gamma[n] * Gamma[n + 5/3] * 3^(n + 1) / (5 * Gamma[2/3] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
Module[{nn=20,pn5},pn5=PolygonalNumber[5,Range[nn]];Table[Det[DiagonalMatrix[Take[pn5,n]]/.(0->1)],{n,nn}]] (* Harvey P. Dale, Feb 07 2025 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-1)/2))); \\ Michel Marcus, Apr 16 2018
  • PARI
    first(n) = my(res = vector(n)); res[1] = 1; for(i = 1, n - 1, res[i + 1] = res[i] * i*(3*i + 5)/2); res \\ David A. Corneth, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = Gamma(n) * Gamma(n + 5/3) * 3^(n + 1) / (5 * Gamma(2/3) * 2^n).
a(n) ~ Gamma(1/3) * 3^(n + 3/2) * n^(2*n + 2/3) / (5 * 2^n * exp(2*n)).
(End)
a(n + 1) = A115067(n + 1) * a(n) = a(n) * n*(3*n + 5)/2. - David A. Corneth, Apr 16 2018

A302910 Determinant of n X n matrix whose main diagonal consists of the first n 6-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 5, 70, 1890, 83160, 5405400, 486486000, 57891834000, 8799558768000, 1663116607152000, 382516819644960000, 105192125402364000000, 34082248630365936000000, 12849007733647957872000000, 5576469356403213716448000000, 2760352331419590789641760000000
Offset: 1

Views

Author

Muniru A Asiru, Apr 15 2018

Keywords

Examples

			The matrix begins:
  1  1  1  1  1  1  1 ...
  1  6  1  1  1  1  1 ...
  1  1 15  1  1  1  1 ...
  1  1  1 28  1  1  1 ...
  1  1  1  1 45  1  1 ...
  1  1  1  1  1 66  1 ...
  1  1  1  1  1  1 91 ...

Crossrefs

Cf. A000384 (hexagonal numbers).
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), this sequence (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).
Odd bisection of column k=1 of A097591.

Programs

  • Maple
    d:=(i,j)->`if`(i<>j,1,i*(2*i-1)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);
  • Mathematica
    nmax = 20; Table[Det[Table[If[i == j, i*(2*i - 1), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[(n + 1/2) * (2*n - 1)! / (3 * 2^(n - 2)), {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[Det[DiagonalMatrix[PolygonalNumber[6,Range[n]]]/.(0->1)],{n,20}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 23 2020 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(2*i-1)))); \\ Michel Marcus, Apr 16 2018

Formula

a(n) = (n + 1/2) * (2*n-1)! / (3 * 2^(n-2)). - Vaclav Kotesovec, Apr 16 2018

A302911 Determinant of n X n matrix whose main diagonal consists of the first n 7-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 6, 102, 3366, 181764, 14541120, 1614064320, 237267455040, 44606281547520, 10437869882119680, 2974792916404108800, 1014404384493801100800, 407790562566508042521600, 190845983281125763900108800, 102865984988526786742158643200
Offset: 1

Views

Author

Muniru A Asiru, Apr 15 2018

Keywords

Examples

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   7   1   1   1   1   1 ...
  1   1  18   1   1   1   1 ...
  1   1   1  34   1   1   1 ...
  1   1   1   1  55   1   1 ...
  1   1   1   1   1  81   1 ...
  1   1   1   1   1   1 112 ...

Crossrefs

Cf. A000566 (heptagonal numbers).
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), this sequence (k=7), A302912 (k=8), A302913 (k=9), A302914 (k=10).

Programs

  • Maple
    d:=(i,j)->`if`(i<>j,1,i*(5*i-3)/2):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..20);
  • Mathematica
    nmax = 20; Table[Det[Table[If[i == j, i*(5*i - 3)/2, 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[5^(n + 1) * Gamma[n] * Gamma[n + 7/5] / (7 * Gamma[2/5] * 2^n)], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(5*i-3)/2))); \\ Michel Marcus, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 5^(n + 1) * Gamma(n) * Gamma(n + 7/5) / (7 * Gamma(2/5) * 2^n).
a(n) ~ Pi * 5^(n+1) * n^(2*n + 2/5) / (7 * Gamma(2/5) * 2^(n-1) * exp(2*n)).
a(n+1) = a(n) * n*(5*n + 7)/2.
(End)

A302912 Determinant of n X n matrix whose main diagonal consists of the first n 8-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 7, 140, 5460, 349440, 33196800, 4381977600, 766846080000, 171773521920000, 47924812615680000, 16294436289331200000, 6631835569757798400000, 3183281073483743232000000, 1779454120077412466688000000, 1145968453329853628547072000000
Offset: 1

Views

Author

Muniru A Asiru, Apr 15 2018

Keywords

Examples

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1   8   1   1   1   1   1 ...
  1   1  21   1   1   1   1 ...
  1   1   1  40   1   1   1 ...
  1   1   1   1  65   1   1 ...
  1   1   1   1   1  96   1 ...
  1   1   1   1   1   1 133 ...

Crossrefs

Cf. A000567 (octagonal numbers).
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), this sequence (k=8), A302913 (k=9), A302914 (k=10).

Programs

  • Maple
    d:=(i,j)->`if`(i<>j,1,i*(3*i-2)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);
  • Mathematica
    nmax = 20; Table[Det[Table[If[i == j, i*(3*i - 2), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[3^(n+1) * Gamma[n] * Gamma[n + 4/3] / (4*Gamma[1/3])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)RecurrenceTable[{a[n+1] == a[n] * n * (3*n + 4), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(3*i-2)))); \\ Michel Marcus, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 3^(n+1) * Gamma(n) * Gamma(n + 4/3) / (4*Gamma(1/3)).
a(n) ~ Pi * 3^(n+1) * n^(2*n + 1/3) / (2 * Gamma(1/3) * exp(2*n)).
a(n+1) = a(n) * n*(3*n + 4).
(End)

A302914 Determinant of n X n matrix whose main diagonal consists of the first n 10-gonal numbers and all other elements are 1's.

Original entry on oeis.org

1, 9, 234, 11934, 1002456, 125307000, 21803418000, 5036589558000, 1490830509168000, 550116457882992000, 247552406047346400000, 133430746859519709600000, 84861955002654535305600000, 62882708656967010661449600000, 53701833193049827104877958400000
Offset: 1

Views

Author

Muniru A Asiru, Apr 15 2018

Keywords

Comments

From Vaclav Kotesovec, Apr 16 2018: (Start)
In general, for k > 2, these determinants for k-gonal numbers satisfies:
a(n,k) = ((k-2)/2)^(n-1) * Gamma(n) * Gamma(n + k/(k-2)) / Gamma(1 + k/(k-2)).
a(n,k) ~ 4*Pi * (k/2 - 1)^n * n^(2*n + 2/(k-2)) / (k * Gamma(k/(k-2)) * exp(2*n)).
a(n+1,k) = a(n,k) * n*((k-2)*n + k)/2.
(End)

Examples

			The matrix begins:
  1   1   1   1   1   1   1 ...
  1  10   1   1   1   1   1 ...
  1   1  27   1   1   1   1 ...
  1   1   1  52   1   1   1 ...
  1   1   1   1  85   1   1 ...
  1   1   1   1   1 126   1 ...
  1   1   1   1   1   1 175 ...

Crossrefs

Cf. A001107.
Cf. Determinant of n X n matrix whose main diagonal consists of the first n k-gonal numbers and all other elements are 1's: A000142 (k=2), A067550 (k=3), A010791 (k=4, with offset 1), A302909 (k=5), A302910 (k=6), A302911 (k=7), A302912 (k=8), A302913 (k=9), this sequence (k=10).
Cf. A007840 (permanent instead of determinant, for k=2).

Programs

  • Maple
    d:=(i,j)->`if`(i<>j,1,i*(4*i-3)):
seq(LinearAlgebra[Determinant](Matrix(n,d)),n=1..16);
  • Mathematica
    nmax = 20; Table[Det[Table[If[i == j, i*(4*i-3), 1], {i, 1, k}, {j, 1, k}]], {k, 1, nmax}] (* Vaclav Kotesovec, Apr 16 2018 *)
RecurrenceTable[{a[n+1] == a[n] * n*(4*n + 5), a[1] == 1}, a, {n, 1, 20}] (* Vaclav Kotesovec, Apr 16 2018 *)
Table[FullSimplify[4^(n+1) * Gamma[n] * Gamma[n + 5/4] / (5*Gamma[1/4])], {n, 1, 15}] (* Vaclav Kotesovec, Apr 16 2018 *)
  • PARI
    a(n) = matdet(matrix(n, n, i, j, if (i!=j, 1, i*(4*i-3)))); \\ Michel Marcus, Apr 16 2018

Formula

From Vaclav Kotesovec, Apr 16 2018: (Start)
a(n) = 4^(n+1) * Gamma(n) * Gamma(n + 5/4) / (5*Gamma(1/4)).
a(n) ~ Pi * 2^(2*n + 3) * n^(2*n + 1/4) / (5 * Gamma(1/4) * exp(2*n)).
a(n+1) = a(n) * n*(4*n + 5).
(End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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