A234576 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.
4, 7, 14, 34, 73, 156, 345, 754, 1640, 3585, 7832, 17091, 37318, 81490, 177913, 388448, 848149, 1851826, 4043232, 8827953, 19274812, 42084287, 91886190, 200622866, 438036729, 956402452, 2088193969, 4559329474, 9954767528, 21735081361, 47456031280
Offset: 4
Examples
For n = 8, a(8) = 34+14+3*7+4 = 73.
Links
- P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.
- P. E. Harris, E. Insko, and L. K. Williams, The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, arXiv preprint arXiv:1401.0055 [math.RT], 2013.
- B. Kostant, A Formula for the Multiplicity of a Weight, Proc Natl Acad Sci U S A. 1958 June; 44(6): 588-589.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
Programs
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Maple
a:=proc(n::nonnegint) if n<=3 then return 0: elif n=4 then return 4: elif n=5 then return 7: elif n=6 then return 14: elif n=7 then return 34: else return a(n-1)+a(n-2)+3*a(n-3)+a(n-4): end if; end proc:
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Mathematica
LinearRecurrence[{1, 1, 3, 1}, {4, 7, 14, 34}, 31] (* Jean-François Alcover, Nov 26 2017 *) -
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
Formula
a(n) = a(n-1) + a(n-2) + 3*a(n-3) + a(n-4).
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