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A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.

%I A292223 #10 Oct 01 2017 00:07:45
%S A292223 60,180,1050,5040,29244,161340,1046250,4825800,27790266,145126548,
%T A292223 843333015,4466836920,26967624184,137243187108,789854179074,
%U A292223 4306147750200,24711052977222,134216193832908,797987818325009,4240082199867228
%N A292223 a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.
%C A292223 This is the sixth column (m = 6) of triangle A213940.
%C A292223 The relevant p(n,6)= A008284(n, 6) representative color multinomials have exponents (signatures) from the six-part partitions of n, written with nonincreasing parts. E.g., n = 8: [3,1,1,1,1,1] and [2,2,1,1,1,1] (p(8,6)=2). The corresponding representative bracelets have the six-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5]*c[6] and c[1]^2*c[2]^2*c[3]*c[4]*c[5]*c[6].
%C A292223 See A056361 for the numbers if also color permutations for D_6 inequivalent bracelets are allowed. (_Andrew Howroyd_ induced me to look at these bracelets.)
%F A292223 a(n) = A213940(n, 6), n >= 6.
%F A292223 a(n) = Sum_{k=b(n, 6)..b(n, 7)-1} A213939(n, k), for n >= 7, with b(n, m) = A214314(n, m) the position where the first m-part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment), and a(6) = A213939(6, b(6,6)) = A213939(6, 11) = 60.
%e A292223 a(6) = A213940(6,6) = A213939(6, 11) = 60 from the representative bracelets (with colors j for c(j), j=1..6) permutations of (1, 2, 3, 4, 5, 6) modulo D_6 (dihedral group) symmetry, i.e., modulo cyclic or anti-cyclic operations. E.g., (1, 2, 3, 4, 6, 5) == (2, 3, 4, 6, 5, 1) == (6, 4, 3, 2, 1, 5) == ..., but (1, 2, 3, 4, 6, 5) is not equivalent to (1, 2, 3, 4, 5, 6). If color permutation is also allowed, then there is only one possibility (see A056361(6) = 1).
%Y A292223 Cf. A008284, A036036, A056361, A213940, A214311, A214314.
%K A292223 nonn
%O A292223 6,1
%A A292223 _Wolfdieter Lang_, Sep 30 2017