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A232162 Number of Weyl group elements, not containing an s_r 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 B and rank n. 3
0, 0, 2, 3, 5, 14, 30, 62, 139, 305, 660, 1444, 3158, 6887, 15037, 32842, 71698, 156538, 341799, 746273, 1629384, 3557592, 7767594, 16959611, 37029365, 80849350, 176525142, 385422198, 841524755, 1837371729, 4011688220, 8759056412 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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=0..31.

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 [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).

FORMULA

a(n) = A232162(n-1) + A232162(n-2) + 3*A232162(n-3) + A232162(n-4).

a(n) = a(n-1)+a(n-2)+3*a(n-3)+a(n-4). G.f.: -x^2*(x+2) / (x^4+3*x^3+x^2+x-1). - Colin Barker, Dec 31 2013

EXAMPLE

For n=4, a(4) = A232162(3) + A232162(2) + 3*A232162(1) + A232162(0) = 3+2+3*0+0=5.

MAPLE

a:=proc(n::nonnegint)

if n=0 then return 0:

elif n=1 then return 0:

elif n=2 then return 2:

elif n=3 then return 3:

else return

a(n-1)+a(n-2)+3*a(n-3)+a(n-4):

end if;

end proc:

MATHEMATICA

LinearRecurrence[{1, 1, 3, 1}, {0, 0, 2, 3}, 32] (* Jean-Fran├žois Alcover, Nov 24 2017 *)

PROG

(PARI) Vec(-x^2*(x+2)/(x^4+3*x^3+x^2+x-1) + O(x^100)) \\ Colin Barker, Dec 31 2013

CROSSREFS

Cf. A232163, A232164, A232165.

Sequence in context: A177901 A143743 A104870 * A243555 A114411 A323768

Adjacent sequences:  A232159 A232160 A232161 * A232163 A232164 A232165

KEYWORD

nonn,easy

AUTHOR

Pamela E Harris, Nov 19 2013

STATUS

approved

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Last modified June 1 17:57 EDT 2020. Contains 334762 sequences. (Running on oeis4.)