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 A196995 Determinant of Killing form K(x,y) of the Lie algebra sl(n,C) for n >=1. 0
 0, -128, -5038848, 140737488355328, 5000000000000000000000000, -354400937492545922690672153504784580608, -72317557999158469111384459491956546088110808312359944192, 57896044618658097711785492504343953926634992332820282019728792003956564819968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS K(x,y) = 2n*Tr(xy) REFERENCES J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972, 21-22 LINKS Table of n, a(n) for n=1..8. FORMULA a(n) = (-1)^binomial(n,2) *2^(n^2-1)*n^(n^2) for n>= 2 MAPLE interface(rtablesize=infinity): with(LinearAlgebra): for n from 1 to 12 do for i from 1 by 1 to n-1 do M[i] := Matrix(n); M[i](i, i) := 1; M[i](i+1, i+1) := -1; end do: ctr := n: for i from 1 by 1 to n do for j from 1 by 1 to n do if(i <> j) then M[ctr] := Matrix(n); M[ctr](i, j) := 1; ctr := ctr +1; end if end do: end do: A := Matrix(n^2-1): for i from 1 by 1 to n^2-1 do for j from 1 by 1 to n^2-1 do A(i, j) := 2*n*Trace(M[i].M[j]): end do: end do: print(Determinant(A)); end do: # Alternatively, using the second description print(0); for n from 2 to 20 do print((-1)^(binomial(n, 2))*2^(n^2-1)*n^(n^2)); end do: CROSSREFS Sequence in context: A016939 A346008 A017011 * A214389 A017095 A017191 Adjacent sequences: A196992 A196993 A196994 * A196996 A196997 A196998 KEYWORD easy,sign AUTHOR Carmen Bruni, Oct 08 2011 STATUS approved

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Last modified September 15 13:44 EDT 2024. Contains 375938 sequences. (Running on oeis4.)