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%I #13 Jan 30 2020 21:29:14
%S 1,4,14,48,167,593,2144,7864,29171,109173,411501,1560089,5943199,
%T 22732739,87253604,335897864,1296447899,5015206349,19439895089,
%U 75487384829,293595204239,1143532045499,4459774977449,17413705988873
%N Number of elements w of the Weyl group D(n) such that the roots sent negative by w span an Abelian subalgebra of the Lie algebra.
%H Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. <a href="https://doi.org/10.1007/s10801-011-0327-z">On the cyclically fully commutative elements of Coxeter groups</a>, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Table 1 FC Type D.
%H C. K. Fan, <a href="http://dx.doi.org/10.1023/A:1022443327568">A Hecke algebra quotient and some combinatorial applications</a>, J. Algebraic Combin. 5 (1996), no. 3, 175-189.
%H C. K. Fan, <a href="http://dx.doi.org/10.1090/S0894-0347-97-00222-1">Structure of a Hecke algebra quotient</a>, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
%H J. R. Stembridge, <a href="http://dx.doi.org/10.1090/S0002-9947-97-01805-9">Some combinatorial aspects of reduced words in finite Coxeter groups</a>, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1285-1332.
%F a(n) = (n+3)*C(n)/2 - 1, where C(n) is a Catalan number (see A000108).
%F D-finite with recurrence: -(n+1)*(3*n^2+n-12)*a(n) +(15*n^3+14*n^2-85*n+36)*a(n-1) -2*(2*n-3)*(3*n^2+7*n-8)*a(n-2)=0. - _R. J. Mathar_, Jun 11 2019
%t Table[(n+3) CatalanNumber[n]/2-1,{n,30}] (* _Harvey P. Dale_, Oct 06 2017 *)
%K nonn,easy
%O 1,2
%A C. Kenneth Fan [ ckfan(AT)MIT.EDU ]
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