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A095922
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Dimension of invariants of n-th tensor power of 5-dimensional irreducible representation of B_2.
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5
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1, 0, 1, 0, 3, 1, 15, 15, 105, 190, 945, 2410, 10263, 31890, 127699, 444458, 1751685, 6518736, 25807445, 100152288, 401449271, 1602902055, 6519160851, 26580508625, 109656966853, 454524861846, 1899821492925, 7982263725826, 33757439931675
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OFFSET
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0,5
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COMMENTS
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The analogous sequence for G_2 is A059710.
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REFERENCES
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Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Springer-Verlag New York (2004).
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LINKS
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FORMULA
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a(n) =sum(A000108(i)*A000108(i+1)*binomial(n, 2*i), i=0..floor(n/2)) - sum(A000108(i)^2*binomial(n, 2*i-1), i=0..floor((n+1)/2)); exponential generating function = exp(t)*b(t) where b(t) is the exponential generating function of the sequence B(n) = (-1)^n*A000108(floor((n+1)/2))*A000108(floor(n/2+1)).
a(0)=1, a(1)=0, a(2)=1 and (n+3)(n+4)a(n)=3(n-1)(n+2)a(n-1)+(n-1)(13n+4)a(n-2)-15(n-1)(n-2)a(n-3) for n>2.
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EXAMPLE
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a(2)=1 because SO(5) has unique (up to multiplication by a constant) invariant in V ⊗ V - the quadratic form x^2+y^2+z^2+u^2+v^2.
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MAPLE
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ca:=n->binomial(n+n, n)/(n+1); a:=n->add(ca(i)*ca(i+1)*binomial(n, 2*i), i=0..floor(n/2))- add(ca(i)^2*binomial(n, 2*i-1), i=0..floor((n+1)/2)); seq(a(n), n=0..40);
A095922:=rsolve({(n+3)*(n+4)*A(n)=3*(n-1)*(n+2)*A(n-1)+(n-1)*(13*n+4)*A(n-2)-15*(n-1)*(n-2)*A(n-3), A(0)=1, A(1)=0, A(2)=1}, A(n), makeproc);
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MATHEMATICA
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t = {0, 1, 0}; Do[AppendTo[ t, (3 (n - 1) (n + 2) t[[n - 1]] + (n - 1) (13 n + 4) t[[n - 2]] - 15 (n - 1) (n - 2) t[[n - 3]])/((n + 3) (n + 4))], {n, 4, 25}]; t = Join[{1}, t] (* T. D. Noe, Apr 11 2014 *)
a[n_] := -n*HypergeometricPFQ[{3/2, 1/2 - n/2, 1 - n/2}, {3, 3}, 16] + HypergeometricPFQ[{3/2, 1/2 - n/2, -n/2}, {2, 3}, 16]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 03 2016 *)
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Alec Mihailovs (alec(AT)mihailovs.com), Jul 11 2004
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STATUS
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approved
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