%I #19 Jun 11 2019 08:10:22
%S 10,42,167,662,2670,10846,44199,180438,737762,3021000,12387990,
%T 50864885,209095841,860447494,3544046278,14608974346,60261567146,
%U 248726602105,1027143932653,4243640251368,17539577253151,72518982292559,299928724501455
%N Number of commutative elements in Coxeter group E_n.
%D C. K. Fan, A Hecke Algebra Quotient and Properties of Commutative Elements of a Weyl Group, MIT Ph.D. Thesis 1995.
%D J. R. Stembridge, Abstracts Amer. Math. Soc., 18 (1) (1997), p. 17, #918-05-495.
%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 Type E.
%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/10.1023/A:1008623323374">The enumeration of fully commutative elements of Coxeter groups</a>, J. Algebraic Combin. 7 (1998), no. 3, 291-320.
%F G.f.: x^3 * ((16-52*x+45*x^2-x^(-1)*(R(x)-1))/(1-7*x+14*x^2-9*x^3) - (6-14*x+12*x^2)/(1-4*x+5*x^2-3*x^3) + (1-x^3-x^4)/(1-x-x^2+x^5)) where R(x)=(1-sqrt(1-4*x))/(2*x) is the generating function for the Catalan numbers. [From Stembridge (1998)] - _Sean A. Irvine_, Sep 01 2015
%o (PARI) x='x+O('x^33); R(x)=(1-sqrt(1-4*x))/(2*x);
%o Vec( x^3 * ((16-52*x+45*x^2-x^(-1)*(R(x)-1))/(1-7*x+14*x^2-9*x^3) - (6-14*x+12*x^2)/(1-4*x+5*x^2-3*x^3) + (1-x^3-x^4)/(1-x-x^2+x^5)) ) \\ _Joerg Arndt_, Sep 02 2015
%Y Cf. A000108.
%K nonn
%O 3,1
%A _Ken Fan_
%E More terms from _Sean A. Irvine_, Sep 01 2015