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%I #40 May 19 2023 11:00:44
%S 0,0,2,3,5,14,30,62,139,305,660,1444,3158,6887,15037,32842,71698,
%T 156538,341799,746273,1629384,3557592,7767594,16959611,37029365,
%U 80849350,176525142,385422198,841524755,1837371729,4011688220,8759056412,19124384574,41755877375,91169119405
%N Number of Weyl group elements, not containing an s_r factor, which contribute 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.
%D P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
%H Stefano Spezia, <a href="/A232162/b232162.txt">Table of n, a(n) for n = 0..2950</a>
%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="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 László Németh and Dragan Stevanović, <a href="https://www.researchgate.net/publication/370765824_Graph_solution_of_system_of_recurrence_equations">Graph solution of system of recurrence equations</a>, Research Gate, 2023. See Table 1 at p. 3.
%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-1) + A232162(n-2) + 3*A232162(n-3) + A232162(n-4).
%F From _Colin Barker_, Dec 31 2013: (Start)
%F a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
%F G.f.: -x^2*(x + 2)/(x^4 + 3*x^3 + x^2 + x - 1). (End)
%e For n=4, a(4) = A232162(3) + A232162(2) + 3*A232162(1) + A232162(0) = 3+2+3*0+0=5.
%p a:=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 a(n-1)+a(n-2)+3*a(n-3)+a(n-4):
%p end if;
%p end proc:
%t LinearRecurrence[{1, 1, 3, 1}, {0, 0, 2, 3}, 32] (* _Jean-François Alcover_, Nov 24 2017 *)
%o (PARI) Vec(-x^2*(x+2)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Dec 31 2013
%Y Cf. A232163, A232164, A232165.
%K nonn,easy
%O 0,3
%A _Pamela E Harris_, Nov 19 2013
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