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%I #12 Sep 07 2018 21:00:21
%S 24,106,464,2003,8560,36333,153584,647775,2729365,11496788,48433965,
%T 204115805,860593940,3630164290,15319869152,64680076487,273183844396,
%U 1154223866418,4878180558021,20622538937234,87202351145432,368810395465291
%N Number of commutative elements in Coxeter group F_n.
%D C. Kenneth Fan, Structure of a Hecke algebra quotient. J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.
%D C. K. Fan, A Hecke algebra quotient and some combinatorial applications. J. Algebraic Combin. 5 (1996), no. 3, 175-189.
%H John R. Stembridge, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=D10.1.1.11.8538&rep=rep1&type=pdf">The Enumeration of Fully Commutative Elements of Coxeter Groups</a>, Dept. of Mathematics, Uni. of Michigan, 1996.
%F From _Sean A. Irvine_, Sep 07 2018: (Start)
%F G.f.: (10 - 5 * (1+x) * (C(x)-1)) / (1-4*x-x^2) + (R(x)-1) / x - (6-4*x) / (1-3*x+x^2) + (1+x) / (1-x-x^2) - 1 / (1-x) where C(x) = (1 - sqrt(1-4*x)) / (2*x) is the g.f. for the Catalan numbers [From Stembridge].
%F a(n) = C(n-1) + 5 * F(3*n-3) - 2 * F(2*n-1) - 2 * F(2*n-3) + F(n) - 1 - 5 * Sum_{k=2..n-1} F(3*k-4) * C(n-k) where C(n) = A000108(n) are the Catalan numbers and F(n) = A000045(n) are the Fibonacci numbers. (End)
%Y Cf. A000045, A000108.
%K nonn
%O 3,1
%A _Ken Fan_
%E More terms from _Sean A. Irvine_, Sep 07 2018
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