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.

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

Offset: 1

Author

Muniru A Asiru, Apr 15 2018

Keywords

• nonn

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

• Muniru A Asiru, Table of n, a(n) for n = 1..100

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