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%I #34 Mar 02 2024 14:19:27
%S 0,1,2,5,10,22,49,106,231,506,1104,2409,5262,11489,25082,54766,119577,
%T 261078,570035,1244610,2717456,5933249,12954570,28284797,61756570,
%U 134838326,294403857,642796690,1403472095,3064318682,6690584704
%N Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra so(2n+1).
%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 B and rank n.
%H P. E. Harris, <a href="https://cpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/d/129/files/2016/04/pamela-e-harris-1zu8lyw.pdf">Combinatorial problems related to Kostant's weight multiplicity formula</a>, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
%H P. E. Harris, E. Insko, and 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 [math.RT], 2013.
%H B. Kostant, <a href="https://doi.org/10.1073/pnas.44.6.588">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#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,3,1).
%F a(n) = A232162(n) + A232162(n-1) + A232162(n-2).
%F a(n) = a(n-1)+a(n-2)+3*a(n-3)+a(n-4). G.f.: -x*(2*x^2+x+1) / (x^4+3*x^3+x^2+x-1). - _Colin Barker_, Jan 01 2014
%e For n=8, a(8) = A232162(8) + A232162(7) + A232162(6) = 139+62+30 = 231.
%p r:=proc(n::nonnegint)
%p if n=0 then return 0:
%p elif n=1 then return 0:
%p elif n=2 then return 2:
%p elif n=3 then return 3:
%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)+r(n-2):
%p end if;
%p end proc:
%t LinearRecurrence[{1, 1, 3, 1}, {0, 1, 2, 5}, 31] (* _Jean-François Alcover_, Nov 26 2017 *)
%o (PARI) Vec(-x*(2*x^2+x+1)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Jan 01 2014
%Y Cf. A232162.
%K nonn,easy
%O 0,3
%A _Pamela E Harris_, Nov 19 2013
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