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A197130 Sum of reflection (or absolute) lengths of all elements in the Coxeter group of type B_n.

%I A197130 #18 Jul 22 2025 12:53:09
%S A197130 1,10,100,1136,14816,220032,3679488,68548608,1409347584,31717048320,
%T A197130 775808778240,20499651624960,582040706088960,17674457139118080,
%U A197130 571655258741145600,19621314364126003200,712374154997583052800,27277192770051951820800
%N A197130 Sum of reflection (or absolute) lengths of all elements in the Coxeter group of type B_n.
%D A197130 P. Renteln, The distance spectra of Cayley graphs of Coxeter groups, Discrete Math., 311 (2011), 738-755.
%H A197130 B. Foster-Greenwood, C. Kriloff, <a href="http://arxiv.org/abs/1502.07392">Spectra of Cayley Graphs of Complex Reflection Groups</a>, arXiv preprint arXiv:1502.07392, 2015
%F A197130 a(n)=Sum_{w in W(B_n)} l_T(w)=|W(B_n)|Sum_{i=1}^n (d_i-1)/d_i=2^n*n!*(1/2+3/4+...+(2n-1)/(2n)) where T=all reflections in W(B_n), l_T(1)=0 and otherwise l_T(w)=min{k|w=t_1*...*t_k for t_i in T}, and d_1,...,d_n are the degrees of W(B_n)
%e A197130 a(2)=10 since W(B_2)={1, t_1=s_1, t_2=s_2, t_3=s_1*s_2*s_1, t_4=s_2*s_1*s_2, t_1*t_2=s_1*s_2, t_2*t_1=s_2*s_1, t_1*t_4=s_1*s_2*s_1*s_2} in terms of simple reflections s_1 and s_2.
%p A197130 seq(2^n*factorial(n)*add((2*k-1)/(2*k),k=1..n),n=1..100);
%t A197130 Table[2^n*Factorial[n]*Sum[(2*k-1)/(2*k),{k,1,n}],{n,1,100}]
%o A197130 (Sage)
%o A197130 [2^n*factorial(n)*sum([(2*k-1)/(2*k) for k in [1..n]]) for n in [1..100]]
%Y A197130 Cf. A067318, A197131.
%K A197130 easy,nonn
%O A197130 1,2
%A A197130 _Cathy Kriloff_, Oct 10 2011