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A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.

%I A214309 #12 Sep 26 2017 14:47:03
%S A214309 3,6,26,93,424,1180,4844,16165,66953,216804,852822,2949804,12119134,
%T A214309 40886724,160826008,572457489,2331396595,8104270828,32043699894,
%U A214309 115995102806,471268872328,1674576998468,6641876380417,24364816845446,98894256728960,357006263815751
%N A214309 a(n) is the number of representative four-color bracelets (necklaces with turning over allowed) with n beads, for n >= 4.
%C A214309 This is the fourth column (m=4) of triangle A213940.
%C A214309 The relevant p(n,4)= A008284(n,4) representative color multinomials have exponents (signatures) from the 4 part partitions of n, written with nonincreasing parts. E.g., n=6: [3,1,1,1] and [2,2,1,1] (p(6,4)=2). The corresponding representative bracelets have the four-color multinomials c[1]^3*c[2]*c[3]*c[4] and c[1]^2*c[2]^2*c[3]*c[4].
%C A214309 Compare this with A032275 where also bracelets with less than four colors are included, and not only representatives are counted.
%C A214309 Number of n-length bracelets w over a 4-ary alphabet {a1,a2,...,a4} such that #(w,a1) >= #(w,a2) >= ... >= #(w,a4) >= 1, where #(w,x) counts the letters x in word w (bracelet analog of A226883). The number of 4 color bracelets up to permutations of colors is given by A056359. - _Andrew Howroyd_, Sep 26 2017
%H A214309 Andrew Howroyd, <a href="/A214309/b214309.txt">Table of n, a(n) for n = 4..200</a>
%F A214309 a(n) = A213940(n,4), n >= 4.
%F A214309 a(n) = sum(A213939(n,k),k=(2+floor(n/2) + p(n,3))..(p(n,4)+1+floor(n/2)+p(n,3))), n>=4, with p(n,m) = A008284(n,m) the number of partitions of n with m parts.
%e A214309 a(4) = A213939(4,5) = 3 from the representative bracelets (with colors  j for c[j], j=1, 2, ..., 4) 1234, 1342 and 1423, all taken cyclically. The necklace cyclic(1324), for example, becomes equivalent to cyclic(1423) under the dihedral D_4 turning over (or reflection) operation.
%e A214309 a(6) = A213939(6, 8) = A213939(6, 9) =  10 + 16 = 26. See the comment above for the representative color multinomials for each case.
%o A214309 (PARI)
%o A214309 Cyc(v)={my(g=fold(gcd,v), s=vecsum(v)); sumdiv(g, d, eulerphi(d)*(s/d)!/prod(i=1, #v, (v[i]/d)!))/s}
%o A214309 CPal(v)={my(odds=#select(t->t%2,v), s=vecsum(v));  if(odds>2, 0, ((s-odds)/2)!/prod(i=1, #v, (v[i]\2)!))}
%o A214309 a(n)={my(t=0); forpart(p=n, t+=Cyc(Vec(p))+CPal(Vec(p)), [1,n], [4,4]); t/2} \\ _Andrew Howroyd_, Sep 26 2017
%Y A214309 Cf. A213939, A213940, A214311 (m=5), A214312 (m=4, all bracelets).
%Y A214309 Cf. A056359, A226883.
%K A214309 nonn
%O A214309 4,1
%A A214309 _Wolfdieter Lang_, Jul 31 2012
%E A214309 Terms a(26) and beyond from _Andrew Howroyd_, Sep 26 2017