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A243094 Cardinality of the Weyl alternation set corresponding to the zero-weight in the representation of the Lie algebra sp(2n) whose highest weight is the second fundamental weight.

%I A243094 #23 Mar 03 2024 18:55:06
%S A243094 1,2,5,8,19,44,92,201,444,965,2104,4602,10045,21924,47879,104540,
%T A243094 228236,498337,1088072,2375657,5186976,11325186,24727205,53988976,
%U A243094 117878715,257374492,561947340,1226946953,2678896484,5849059949,12770744632,27883440986,60880261949
%N A243094 Cardinality of the Weyl alternation set corresponding to the zero-weight in the representation of the Lie algebra sp(2n) whose highest weight is the second fundamental weight.
%C A243094 Number of Weyl group elements contributing nonzero terms to Kostant's weight multiplicity formula when computing the multiplicity of the zero-weight in the defining representation for the Lie algebra of type C and rank n. Here the highest weight would be the second fundamental weight of sp(2n).
%H A243094 P. E. Harris, <a href="https://people.uwm.edu/jw/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 A243094 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 [math.RT], 2013.
%H A243094 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 A243094 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 3, 1).
%F A243094 a(n) = A232162(n) + A232162(n-1).
%F A243094 a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
%F A243094 G.f.: (x^4 + 2*x^3 - 2*x^2 - x - 1) / (x^4 + 3*x^3 + x^2 + x - 1). - _Joerg Arndt_, Aug 18 2014
%p A243094 r:=proc(n::nonnegint) option remember
%p A243094 if n=0 then return 0:
%p A243094 elif n=1 then return 0:
%p A243094 elif n=2 then return 2:
%p A243094 elif n=3 then return 3:
%p A243094 else return
%p A243094 r(n-1)+r(n-2)+3*r(n-3)+r(n-4):
%p A243094 end if;
%p A243094 end proc:
%p A243094 a:=proc(n::nonnegint)
%p A243094 if n=0 then return 0:
%p A243094 elif n=1 then return 1:
%p A243094 else return
%p A243094 r(n)+r(n-1):
%p A243094 end if;
%p A243094 end proc:
%t A243094 Join[{1}, LinearRecurrence[{1, 1, 3, 1}, {2, 5, 8, 19}, 32]] (* _Jean-François Alcover_, Dec 05 2017 *)
%o A243094 (PARI) Vec( (x^4+2*x^3-2*x^2-x-1) / (x^4+3*x^3+x^2+x-1) +O(x^66) ) \\ _Joerg Arndt_, Aug 18 2014
%Y A243094 Cf. A232162.
%K A243094 nonn
%O A243094 0,2
%A A243094 _Pamela E Harris_, Aug 18 2014