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%I #8 Mar 31 2012 10:31:29
%S 0,-128,-5038848,140737488355328,5000000000000000000000000,
%T -354400937492545922690672153504784580608,
%U -72317557999158469111384459491956546088110808312359944192,57896044618658097711785492504343953926634992332820282019728792003956564819968
%N Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1.
%C K(x,y) = 2n*Tr(xy)
%D J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22
%F a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2
%p interface(rtablesize=infinity):
%p with(LinearAlgebra):
%p for n from 1 to 12 do
%p for i from 1 by 1 to n-1 do
%p M[i] := Matrix(n);
%p M[i](i,i) := 1;
%p M[i](i+1,i+1) := -1;
%p end do:
%p ctr := n:
%p for i from 1 by 1 to n do
%p for j from 1 by 1 to n do
%p if(i <> j) then
%p M[ctr] := Matrix(n);
%p M[ctr](i,j) := 1;
%p ctr := ctr +1;
%p end if
%p end do:
%p end do:
%p A := Matrix(n^2-1):
%p for i from 1 by 1 to n^2-1 do
%p for j from 1 by 1 to n^2-1 do
%p A(i,j) := 2*n*Trace(M[i].M[j]):
%p end do:
%p end do:
%p print(Determinant(A));
%p end do:
%p # Alternatively, using the second description
%p print(0);
%p for n from 2 to 20 do
%p print((-1)^(binomial(n,2))*2^(n^2-1)*n^(n^2));
%p end do:
%K easy,sign
%O 1,2
%A _Carmen Bruni_, Oct 08 2011
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