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%I #22 Oct 08 2017 19:49:53
%S 1,13,135,1537,19811,289073,4741923,86705417,1752264235,38832482641,
%T 937035652035,24465531961465,687363659349179,20679220894484897,
%U 663327190230305715,22600083539456536457,815088161465498630635
%N Number of elements of a Weyl group of order 2^{n-1} n! of type D for which a reduced word contains all of the simple reflections.
%C This is an analog for type D of the concept of connected permutations (see A003319 and A109253).
%H Vaclav Kotesovec, <a href="/A112225/b112225.txt">Table of n, a(n) for n = 2..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 G.f.: f(x) = (g(2x)+3)/(2 g(x)) + x - 2 where g(x) = sum_{n >= 0} n! x^n.
%F a(n) ~ n! * 2^(n-1) * (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 29 2015
%e For n=2, the Weyl group of order 4 is generated by {s_0', s_1} with (s_0')^2=s_1^2 = (s_0' s_1)^2 = 1, s_0' s_1 is the only element with a reduced word containing both simple reflections (the other elements are 1, s_0' and s_1).
%e For n=3, the Weyl group of type D is isomorphic to S_4 where there are 13 'connected permutations' (see A003319).
%p f:=n->coeff(series((add(2^k*k!*x^k,k=1..n)+4)/add(2*k!*x^k,k=0..n)+x-2,x,n+1),x,n);
%t nmax = 20; Rest[Rest[CoefficientList[Assuming[Element[x, Reals], Series[(Exp[1/(2*x)] * ExpIntegralEi[1/(2*x)] + 6*x*Exp[1/x]) / (4*ExpIntegralEi[1/x]) + x - 2, {x, 0, nmax}]], x]]] (* _Vaclav Kotesovec_, Aug 05 2015 after Martin and Kearney *)
%Y Cf. A003319, A109253, A260952.
%K nonn
%O 2,2
%A _Mike Zabrocki_, Aug 28 2005
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