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A232165
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Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra sp(2n).
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2
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0, 1, 2, 3, 8, 18, 37, 82, 181, 392, 856, 1873, 4086, 8919, 19480, 42530, 92853, 202742, 442665, 966496, 2110240, 4607473, 10059866, 21964555, 47957080, 104708706, 228619317, 499163818, 1089866333, 2379596808, 5195573912, 11343933537, 24768164206, 54078416287
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OFFSET
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0,3
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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 adjoint representation for the Lie algebra of type C and rank n.
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REFERENCES
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P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
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LINKS
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Table of n, a(n) for n=0..33.
P. E. Harris, E. Insko, L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055, 2013
B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
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FORMULA
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a(n) = A232164(n) + A232164(n-1).
a(n) = a(n-1)+a(n-2)+3*a(n-3)+a(n-4). G.f.: -x*(x+1) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Jan 01 2014
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EXAMPLE
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For n=3, a(3) = A232164(3) + A232164(2) = 2+1 = 3.
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MAPLE
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r:=proc(n::nonnegint)
if n=0 then return 0:
elif n=1 then return 1:
elif n=2 then return 1:
elif n=3 then return 2:
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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LinearRecurrence[{1, 1, 3, 1}, {0, 1, 2, 3}, 40] (* Harvey P. Dale, Nov 22 2014 *)
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PROG
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(PARI) Vec(-x*(x+1)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Jan 01 2014
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CROSSREFS
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Cf. A232164.
Sequence in context: A096254 A091765 A036746 * A129955 A034066 A034076
Adjacent sequences: A232162 A232163 A232164 * A232166 A232167 A232168
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KEYWORD
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nonn,easy
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AUTHOR
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Pamela E Harris, Nov 19 2013
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STATUS
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approved
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