%I A096364
%S A096364 0,1,1,5,41,459,6469,109577,2164273,48787127,1235194181,34688329389,
%T A096364 1069808023129,35936710441475,1305872544724357,51034409943693329,
%U A096364 2134268774190839009,95096941799140816623,4497325804679310925957
%N A096364 Number of ways to generate a Coxeter element of the reflection group 
               of the root system B_n with certain restrictions on generators: (3n-4)*(n-2)^(n-2) 
               - (n-1)^(n-1).
%C A096364 Let G be the group (Z/2Z)^n * S_n. (Here * denotes the semidirect product. 
               Each member of G is a function f from {-n, -n+1, ..., -2, -1, 1, 
               2, ..., n-1, n} to itself such that f is bijective and f(-x) = -f(x). 
               The operation is composition.) Let C be a Coxeter element of G such 
               that C(i) = i+1 for i = 1, 2, ..., n-1 and C(n) = -1. Let s_ij, i 
               eq j and i,j > 0, be the function such that s_ij(i) = j, s_ij(j) 
               = i and s_ij(k) = k for all k eq i,j, k > 0. Let s_2 be the function 
               such that s_2(2) = -2 and s_2(k) = k for all k eq 2. Then a(n) is 
               the number of ways to write C as a product of the s_ij's and s_2, 
               using as few elements as possible.
%C A096364 a(n) is the number of ways to write the Coxeter element s_{e_1}s_{e_1-e_2}s_{e_2-e_3}s_{e_3-e_4}...s_{e_{n-1}\
               -e_n} of the reflection group of the root system B_n as a product 
               of reflections from {s_{e_i - e_j}, e_2}, using as few reflections 
               as possible.
%C A096364 a(n) is also the number of ways to write the same Coxeter element as 
               a product of reflections from {s_{e_i - e_j}, e_2} - {s_{e1-e2}} 
               \union {s_{e1+e2}}, using as few reflections as possible.
%C A096364 Let T be a spanning tree in complete graph K_n on n labeled nodes. Let 
               w(T) be the least neighbor of n in T. Let A_n be the set of spanning 
               tree T of K_n such that the least neighbor or n is a leaf. Then for 
               n >= 3, sum_{T \in A_n} w(T) = a(n).
%D A096364 Richard Kane, Reflection Groups and Invariant Theory, Springer-Verlag, 
               New York, 2001; 9-10.
%F A096364 a(n) = (3n-4)*(n-2)^(n-2) - (n-1)^(n-1) for n >= 3.
%F A096364 E.g.f.: (x*(x-1)*LambertW(-x)-x^2)/LambertW(-x)^2. - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jul 02 2004
%e A096364 a(4)=5 because we can write s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4} =
%e A096364 1) s_{e1-e2}s_{e2}s_{e2-e3}s_{e3-e4}
%e A096364 2) s_{e1-e2}s_{e2}s_{e2-e4}s_{e2-e3}
%e A096364 3) s_{e1-e2}s_{e2}s_{e3-e4}s_{e2-e4}
%e A096364 4) s_{e1-e2}s_{e3-e4}s_{e2}s_{e2-e4}
%e A096364 5) s_{e3-e4}s_{e1-e2}s_{e2}s_{e2-e4}
%t A096364 a[1] = 0; a[2] = a[3]1; a[n_] := (3n - 4)*(n - 2)^(n - 2) - (n - 1)^(n 
               - 1); Table[ a[n], {n, 19}] (from Robert G. Wilson v Jul 24 2004)
%Y A096364 Sequence in context: A083073 A115257 A047735 this_sequence A049119 A032188 
               A143415
%Y A096364 Adjacent sequences: A096361 A096362 A096363 this_sequence A096365 A096366 
               A096367
%K A096364 nonn
%O A096364 1,4
%A A096364 Pramook Khungurn (pramook(AT)mit.edu), Jul 01 2004
%E A096364 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 24 2004