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%I #24 Feb 10 2024 03:46:34
%S 4,7,14,34,73,156,345,754,1640,3585,7832,17091,37318,81490,177913,
%T 388448,848149,1851826,4043232,8827953,19274812,42084287,91886190,
%U 200622866,438036729,956402452,2088193969,4559329474,9954767528,21735081361,47456031280
%N Number of Weyl group elements, not containing s_1 or s_2, 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 D 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) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
%F G.f.: -x^4*(x^3+3*x^2+3*x+4) / (x^4+3*x^3+x^2+x-1). - _Colin Barker_, Dec 30 2013
%e For n = 8, a(8) = 34+14+3*7+4 = 73.
%p a:=proc(n::nonnegint)
%p if n<=3 then return 0:
%p elif n=4 then return 4:
%p elif n=5 then return 7:
%p elif n=6 then return 14:
%p elif n=7 then return 34:
%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}, {4, 7, 14, 34}, 31] (* _Jean-François Alcover_, Nov 26 2017 *)
%o (PARI) Vec(-x^4*(x^3+3*x^2+3*x+4)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ _Colin Barker_, Dec 30 2013
%K nonn,easy
%O 4,1
%A _Erik Insko_, Dec 28 2013
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