A118714 Determinant of n X n matrix whose diagonal contains the first n tetrahedral numbers and all other elements are 1's.

1, 3, 27, 513, 17442, 959310, 79622730, 9475104870, 1553917198680, 340307866510920, 96987741955612200, 35206550329887228600, 15983773849768801784400, 8934929582020760197479600, 6066817186192096174088648400, 4944456006746558381882248446000

Offset: 1

Author

Alexander Adamchuk, May 20 2006

Keywords

• nonn

Comments

a(n+2) / a(n+1) = A062748(n) = A062745(n+2, 3)= binomial(n+4, 3)-1 = (n+1)*(n^2+8*n+18)/3!.

Examples

			The matrix begins:
1 1 1 1 1 1 1 ...
1 4 1 1 1 1 1 ...
1 1 10 1 1 1 1 ...
1 1 1 20 1 1 1 ...
1 1 1 1 35 1 1 ...
1 1 1 1 1 56 1 ...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..150

Crossrefs

Cf. A000292, A067550, A062748, A062745.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 1,
      a(n-1) *(6+4*n+n^2)*(n-1)/6)
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 15 2015

Mathematica

Table[ Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ], {n, 1, 20} ]
Table[Product[(k-3)*(k^2+2)/3!,{k,4,n+2}],{n,1,20}]

PARI

a(n) = matdet(matrix(n, n, i, j, if(i==j, i*(i+1)*(i+2)/6, 1))) \\ Colin Barker, Nov 13 2015

Formula

a(n) = Det[ DiagonalMatrix[ Table[ i*(i+1)(i+2)/6 - 1, {i, 1, n} ] ] + 1 ].

a(n) = Product[(j-3)*(j^2+2)/3!,{j,4,n+2}].

a(n) = Product[(k+1)*(k^2+8*k+18)/3!,{k,0,n-2}] = Product[A062748(k),{k,0,n-2}].

a(n) ~ sqrt(Pi) * sinh(Pi*sqrt(2)) * n^(3*n + 9/2) / (11 * 2^(n-1) * 3^(n+1) * exp(3*n)). - Vaclav Kotesovec, Apr 17 2018

Extensions

a(15) and a(16) from Colin Barker, Nov 13 2015