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%I #27 Aug 05 2015 08:42:18
%S 1,1,5,35,309,3287,41005,588487,9571125,174230863,3513016445,
%T 77760961991,1875249535941,48946667107295,1374949148971597,
%U 41361812577803383,1326708910645563669,45201102932347559503,1630193308027321807133,62047171055048539457255
%N Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections.
%C This is the analog of a connected permutation (permutation with no global ascent) in type B.
%H Vaclav Kotesovec, <a href="/A109253/b109253.txt">Table of n, a(n) for n = 0..400</a>
%H N. Bergeron, C. Hohlweg, M. Zabrocki, <a href="http://arXiv.org/abs/math.CO/0509271">Posets related to the connectivity set of Coxeter groups</a>, arXiv:math/0509271 [math.CO], 2005-2006.
%H Richard J. Martin, and Michael J. Kearney, <a href="http://dx.doi.org/10.1007/s00493-014-3183-3">Integral representation of certain combinatorial recurrences</a>, Combinatorica: 35:3 (2015), 309-315.
%F O.g.f.: g(2x)/g(x) where g(x) = sum_{n>=0} n! x^n.
%F a(n) ~ n! * 2^n * (1 - 1/(2*n) - 1/(4*n^2) - 5/(8*n^3) - 35/(16*n^4) - 319/(32*n^5) - 3557/(64*n^6) - 46617/(128*n^7) - 699547/(256*n^8) - 11801263/(512*n^9) - 220778973/(1024*n^10)), for coefficients see A260952. - _Vaclav Kotesovec_, Jul 28 2015
%e For n=2, the Weyl group B_2 has 8 elements and is generated by {t,s} with s^2=t^2=(st)^4=1, the elements which have reduced words containing both s and t are st, ts, sts, tst and stst. The other three elements are 1, s, t. Therefore f(2)=5.
%p f:=k->coeff(series(add(2^n*n!*x^n,n=0..k)/add(n!*x^n,n=0..k),x,k+1),x,k);
%t nmax = 20; CoefficientList[Assuming[Element[x, Reals], Series[1/2*Exp[1/(2*x)] * ExpIntegralEi[1/(2*x)] / ExpIntegralEi[1/x], {x, 0, nmax}]], x] (* _Vaclav Kotesovec_, Aug 05 2015 *)
%Y Cf. A003319, A109281, A112225, A260952.
%K nonn
%O 0,3
%A _Mike Zabrocki_, Aug 19 2005
%E More terms from _Vaclav Kotesovec_, Aug 05 2015
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