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A243094
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Cardinality of the Weyl alternation set corresponding to the zero-weight in the representation of the Lie algebra sp(2n) whose highest weight is the second fundamental weight.
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0
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1, 2, 5, 8, 19, 44, 92, 201, 444, 965, 2104, 4602, 10045, 21924, 47879, 104540, 228236, 498337, 1088072, 2375657, 5186976, 11325186, 24727205, 53988976, 117878715, 257374492, 561947340, 1226946953, 2678896484, 5849059949, 12770744632, 27883440986, 60880261949
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OFFSET
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0,2
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COMMENTS
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Number of Weyl group elements contributing nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the defining representation for the Lie algebra of type C and rank n. Here the highest weight would be the second fundamental weight of sp(2n).
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
G.f.: (x^4 + 2*x^3 - 2*x^2 - x - 1) / (x^4 + 3*x^3 + x^2 + x - 1). - Joerg Arndt, Aug 18 2014
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MAPLE
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r:=proc(n::nonnegint) option remember
if n=0 then return 0:
elif n=1 then return 0:
elif n=2 then return 2:
elif n=3 then return 3:
else return
r(n-1)+r(n-2)+3*r(n-3)+r(n-4):
end if;
end proc:
a:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
else return
r(n)+r(n-1):
end if;
end proc:
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MATHEMATICA
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Join[{1}, LinearRecurrence[{1, 1, 3, 1}, {2, 5, 8, 19}, 32]] (* Jean-François Alcover, Dec 05 2017 *)
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PROG
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(PARI) Vec( (x^4+2*x^3-2*x^2-x-1) / (x^4+3*x^3+x^2+x-1) +O(x^66) ) \\ Joerg Arndt, Aug 18 2014
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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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