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A056345 Number of bracelets of length n using exactly five different colored beads.

%I A056345 #27 Oct 13 2018 09:30:18
%S A056345 0,0,0,0,12,150,1200,7905,46400,255636,1346700,6901725,34663020,
%T A056345 171786450,843130688,4110958530,19951305240,96528492700,466073976900,
%U A056345 2247627076731,10832193571460,52194109216950
%N A056345 Number of bracelets of length n using exactly five different colored beads.
%C A056345 Turning over will not create a new bracelet.
%D A056345 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%F A056345 a(n) = A032276(n) - 5*A032275(n) + 10*A027671(n) - 10*A000029(n) + 5.
%F A056345 From _Robert A. Russell_, Sep 27 2018: (Start)
%F A056345 a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=5 is the number of colors and S2 is the Stirling subset number A008277.
%F A056345 G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=5 is the number of colors. (End)
%e A056345 For a(5)=12, pair up the 24 permutations of BCDE, each with its reverse, such as BCDE-EDCB.  Precede the first of each pair with an A, such as ABCDE.  These are the 12 arrangements, all chiral.  If we precede the second of each pair with an A, such as AEDCB, we get the chiral partner of each. - _Robert A. Russell_, Sep 27 2018
%t A056345 t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
%t A056345 T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
%t A056345 a[n_] := T[n, 5];
%t A056345 Array[a, 22] (* _Jean-François Alcover_, Nov 05 2017, after _Andrew Howroyd_ *)
%t A056345 k=5; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* _Robert A. Russell_, Sep 27 2018 *)
%Y A056345 Column 5 of A273891.
%Y A056345 Equals (A056285 + A056491) / 2 = A056285 - A305544 = A305544 + A056491.
%K A056345 nonn
%O A056345 1,5
%A A056345 _Marks R. Nester_