A158046 Determinant of power series with alternate signs of gamma matrix with determinant 3!.

6, -12, 294, -4800, 33006, -868476, 8045022, -133497600, 1840843662, -23069939772, 357884304366, -4506695659200, 65700186820638, -892588899692796, 12240418932523614, -172125321194572800, 2335747604463776238, -32681605781959208508, 448728077274231515214

Offset: 1

Author

Giorgio Balzarotti and Paolo P. Lava, Mar 11 2009

Keywords

• sign

Comments

a(n) = Determinant(A - A^2 + A^3 - A^4 + A^5 - ... - (-1)^n*A^n).

where A is the submatrix A(1..4,1..4) of the matrix with factorial determinant

A = [[1,1,1,1,1,1,...], [1,2,1,2,1,2,...], [1,2,3,1,2,3,...], [1,2,3,4,1,2,...], [1,2,3,4,5,1,...], [1,2,3,4,5,6,...], ...]; note: Determinant A(1..n,1..n) = (n-1)!.

a(n) is even with respect to signs of power of A.

Examples

			a(1) = Determinant(A) = 3! = 6.
		

References

• G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008

Crossrefs

Cf. A111490, A158040-A158045.

Programs

Maple

with(LinearAlgebra):
A:= Matrix([[1, 1, 1, 1], [1, 2, 1, 2], [1, 2, 3, 1], [1, 2, 3, 4]]):
a:= n-> Determinant(add(A^i*(-1)^(i-1), i=1..n)):
seq(a(n), n=1..30);

PARI

vector(100, n, matdet(sum(k=1, n, [1,1,1,1 ; 1,2,1,2 ; 1,2,3,1 ; 1,2,3,4]^k*(-1)^(k-1)))) \\ Colin Barker, Jul 14 2014

Formula

Empirical g.f.: -6*x*(6*x^2 -1)*(46656*x^12 -190512*x^10 +60480*x^9 +243432*x^8 -21168*x^7 -100984*x^6 -3528*x^5 +6762*x^4 +280*x^3 -147*x^2 +1) / ((x -1)*(6*x -1)*(6*x^4 +22*x^3 +23*x^2 +10*x +1)*(216*x^4 +360*x^3 +138*x^2 +22*x +1)*(216*x^6 -828*x^5 +1284*x^4 -808*x^3 +214*x^2 -23*x +1)). - Colin Barker, Jul 14 2014

Extensions

More terms, and offset changed to 1 by Colin Barker, Jul 14 2014