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A247591
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Dimension of invariants of 2n-th tensor power of 6-dimensional irreducible representation of A_3.
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0
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1, 1, 3, 16, 126, 1296, 16071, 228514, 3607890, 61891050, 1135871490, 22049362440, 448790912004, 9512960347260, 208858963314735, 4728736078065810, 110006925920592810, 2621619942885055530, 63840054782606886630, 1585094577104979776880, 40054740803371374834780, 1028483346608802276173280
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OFFSET
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0,3
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COMMENTS
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The 6-dimensional representation is the usual representation of SO(6)=A_3.
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LINKS
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FORMULA
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a(n) = (-36*(n-1)*(2*n-3)*(2*n-1)*a(n-2) + 4*n*(2*n-1)*(5*n+7)*a(n-1)) / ((n+2)*(n+3)^2).
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EXAMPLE
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For 2n=6, there are 15 invariants corresponding to all ways of pairing the 6 indices with the metric tensor, plus one invariant which is the completely skew-symmetric 6-index tensor.
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MATHEMATICA
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a[0] = 1; a[1] = 1; a[n_] := a[n] = (4*n*(2*n-1)*(5*n+7)*a[n-1] - 36*(n-1)*(2*n-3)*(2*n-1)*a[n-2]) / ((n+2)*(n+3)^2); Table[a[n], {n, 0, 21}]
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PROG
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(PARI) N=66; v=vector(N); v[1]=1; v[2]=1;
for(n=2, N-1, my(t=n+1); v[t] = (-36*(n-1)*(2*n-3)*(2*n-1)*v[t-2] + 4*n*(2*n-1)*(5*n+7)*v[t-1]) / ((n+2)*(n+3)^2) );
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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