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A232165 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra sp(2n). 2

%I

%S 0,1,2,3,8,18,37,82,181,392,856,1873,4086,8919,19480,42530,92853,

%T 202742,442665,966496,2110240,4607473,10059866,21964555,47957080,

%U 104708706,228619317,499163818,1089866333,2379596808,5195573912,11343933537,24768164206,54078416287

%N Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra sp(2n).

%C 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.

%D P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

%H P. E. Harris, E. Insko, L. K. Williams, <a href="http://arxiv.org/abs/1401.0055">The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula</a>, arXiv preprint arXiv:1401.0055, 2013

%H B. Kostant, <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC528626/">A Formula for the Multiplicity of a Weight</a>, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,3,1).

%F a(n) = A232164(n) + A232164(n-1).

%F 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

%e For n=3, a(3) = A232164(3) + A232164(2) = 2+1 = 3.

%p r:=proc(n::nonnegint)

%p if n=0 then return 0:

%p elif n=1 then return 1:

%p elif n=2 then return 1:

%p elif n=3 then return 2:

%p else return

%p r(n-1)+r(n-2)+3*r(n-3)+r(n-4):

%p end if;

%p end proc:

%p a:=proc(n::nonnegint)

%p if n=0 then return 0:

%p elif n=1 then return 1:

%p else return

%p r(n)+r(n-1):

%p end if;

%p end proc:

%t LinearRecurrence[{1,1,3,1},{0,1,2,3},40] (* _Harvey P. Dale_, Nov 22 2014 *)

%o (PARI) Vec(-x*(x+1)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Jan 01 2014

%Y Cf. A232164.

%K nonn,easy

%O 0,3

%A _Pamela E Harris_, Nov 19 2013

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Last modified June 4 17:50 EDT 2020. Contains 334828 sequences. (Running on oeis4.)