OFFSET
1,1
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.
REFERENCES
G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008
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
EXAMPLE
a(1) = Determinant(A) = 3! = 6.
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);
PROG
(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
CROSSREFS
KEYWORD
sign
AUTHOR
Giorgio Balzarotti and Paolo P. Lava, Mar 11 2009
EXTENSIONS
More terms, and offset changed to 1 by Colin Barker, Jul 14 2014
STATUS
approved