A197131 Sum of the reflection (absolute) lengths of all elements in the Coxeter group of type D_n.
4, 46, 544, 7216, 108096, 1816704, 33951744, 699512832, 15765626880, 386046443520, 10208951009280, 290039357767680, 8811722692362240, 285113464809062400, 9789232245217689600, 355501479519741542400, 13615286053738276454400, 548476851979845579571200
Offset: 2
Examples
a(3)=46 because W(D_3)=W(A_3) and in sequence A067318, a(3)=46.
References
- P. Renteln, The distance spectra of Cayley graphs of Coxeter groups, Discrete Math., 311 (2011), 738-755.
Links
- B. Foster-Greenwood, C. Kriloff, Spectra of Cayley Graphs of Complex Reflection Groups, arXiv preprint arXiv:1502.07392, 2015
Programs
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Maple
seq(2^(n-1)*factorial(n)*(add((2*k-1)/(2*k), k=1..n-1)+(n-1)/n), n=2..100);
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Mathematica
Table[2^(n-1)*Factorial[n]*(Sum[(2*k-1)/(2*k),{k,1,n-1}]+(n-1)/n), {n,2,100}] -
Sage
[2^(n-1)*factorial(n)*(sum([(2*k-1)/(2*k) for k in [1..n-1]])+(n-1)/n) for n in [2..100]]
Formula
a(n)=Sum_{w in W(D_n)} l_T(w)=|W(D_n)|Sum_{i=1}^n (d_i-1)/d_i=2^(n-1)*n!*(1/2+3/4+...+(2n-3)/(2n-2)+(n-1)/n) where T=all reflections in W(D_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(D_n)