A234597 Number of Weyl group elements, not containing an s_1 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 D and rank n.
5, 11, 21, 48, 107, 229, 501, 1099, 2394, 5225, 11417, 24923, 54409, 118808, 259403, 566361, 1236597, 2699975, 5895058, 12871185, 28102765, 61359099, 133970477, 292509056, 638659595, 1394439181, 3044596421, 6647523443, 14514097002, 31689848889, 69191112641
Offset: 4
Examples
For n=6, a(6) = A234576(6) + A234576(5)= 14+7 = 21.
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, 2013
- B. Kostant, A Formula for the Multiplicity of a Weight, Proc. Natl. Acad. Sci. USA, 44 (No. 6, June 1958), 588-589.
- Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).
Crossrefs
Cf. A234576.
Programs
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Maple
r:=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 r(n-1)+r(n-2)+3*r(n-3)+r(n-4): end if; end proc: a:=proc(n::nonnegint) if n<=3 then return 0: elif n=4 then return 5: elif n=5 then return 11: else return r(n)+r(n-1): end if; end proc:
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Mathematica
LinearRecurrence[{1,1,3,1},{5,11,21,48},40] (* Harvey P. Dale, Feb 17 2016 *) -
PARI
Vec(-x^4*(x^3+5*x^2+6*x+5)/(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+5*x^2+6*x+5) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Dec 30 2013