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A234598 Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra of so(2n). 0
9, 18, 35, 82, 180, 385, 846, 1853, 4034, 8810, 19249, 42014, 91727, 200298, 437316, 954809, 2084746, 4551801, 9938290, 21699138, 47377577, 103443386, 225856667, 493131922, 1076696324, 2350841633, 5132790390, 11206852917, 24468864530 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,1

COMMENTS

Cardinality of the Weyl alternation set corresponding to the zero-weight in the adjoint representation of the Lie algebra of type D and rank n.

REFERENCES

P. E. Harris, Combinatorial problems related to Kostant's weight multiplicity formula, PhD Dissertation, University of Wisconsin-Milwaukee, 2012.

LINKS

Table of n, a(n) for n=4..32.

P. E. Harris, E. Insko, 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 U S A. 1958 June; 44(6): 588-589.

Index entries for linear recurrences with constant coefficients, signature (1,1,3,1).

FORMULA

a(n) = A234597(n) + A234576(n).

a(n) = 3*A234576(n-1) + 2*A234576(n-2) + 6*A234576(n-3) + 2*A234576(n-4).

G.f.: x^4*(2*x^3 + 8*x^2 + 9*x + 9)/(-x^4 - 3*x^3 - x^2 - x + 1). - Ralf Stephan, Jan 05 2014

EXAMPLE

For n = 8, a(n) = 107+73 = 180 and a(n) = 3(34) + 2(14) + 6(7) + 2(4) = 180.

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 9:

elif n=5 then return 18:

elif n=6 then return 35:

elif n=5 then return 82:

else return

3*r(n-1)+2*r(n-2)+6*r(n-3)+2*r(n-4):

end if;

end proc:

MATHEMATICA

LinearRecurrence[{1, 1, 3, 1}, {9, 18, 35, 82}, 30] (* Jean-Fran├žois Alcover, Dec 06 2017 *)

CROSSREFS

Cf. A232163, A232165, A234576, A234597.

Sequence in context: A139591 A191268 A109668 * A051063 A173942 A162689

Adjacent sequences:  A234595 A234596 A234597 * A234599 A234600 A234601

KEYWORD

nonn,easy

AUTHOR

Erik Insko, Dec 28 2013

EXTENSIONS

More terms from Ralf Stephan, Jan 05 2014

STATUS

approved

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Last modified January 27 04:57 EST 2020. Contains 331291 sequences. (Running on oeis4.)