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

%I A158046 #16 Nov 27 2015 00:35:22
%S A158046 6,-12,294,-4800,33006,-868476,8045022,-133497600,1840843662,
%T A158046 -23069939772,357884304366,-4506695659200,65700186820638,
%U A158046 -892588899692796,12240418932523614,-172125321194572800,2335747604463776238,-32681605781959208508,448728077274231515214
%N A158046 Determinant of power series with alternate signs of gamma matrix with determinant 3!.
%C A158046 a(n) = Determinant(A - A^2 + A^3 - A^4 + A^5 - ... - (-1)^n*A^n).
%C A158046 where A is the submatrix A(1..4,1..4) of the matrix with factorial determinant
%C A158046 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)!.
%C A158046 a(n) is even with respect to signs of power of A.
%D A158046 G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008
%F A158046 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
%e A158046 a(1) = Determinant(A) = 3! = 6.
%p A158046 with(LinearAlgebra):
%p A158046 A:= Matrix([[1, 1, 1, 1], [1, 2, 1, 2], [1, 2, 3, 1], [1, 2, 3, 4]]):
%p A158046 a:= n-> Determinant(add(A^i*(-1)^(i-1), i=1..n)):
%p A158046 seq(a(n), n=1..30);
%o A158046 (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
%Y A158046 Cf. A111490, A158040-A158045.
%K A158046 sign
%O A158046 1,1
%A A158046 _Giorgio Balzarotti_ and _Paolo P. Lava_, Mar 11 2009
%E A158046 More terms, and offset changed to 1 by _Colin Barker_, Jul 14 2014