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A056344 Number of bracelets of length n using exactly four different colored beads.

%I A056344 #31 Oct 14 2018 04:12:44
%S A056344 0,0,0,3,24,136,612,2619,10480,41388,159780,614058,2341920,8919816,
%T A056344 33905188,128907279,490213680,1866127840,7111777860,27140369148,
%U A056344 103721218000,396974781456,1521577377012,5840547488954
%N A056344 Number of bracelets of length n using exactly four different colored beads.
%C A056344 Turning over will not create a new bracelet.
%D A056344 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 A056344 a(n) = A032275(n) - 4*A027671(n) + 6*A000029(n) - 4.
%F A056344 From _Robert A. Russell_, Sep 27 2018: (Start)
%F A056344 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=4 is the number of colors and S2 is the Stirling subset number A008277.
%F A056344 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=4 is the number of colors.
%F A056344 a(n) = (A056284(n) + A056490(n)) / 2 = A056284(n) - A305543(n) = A305543(n) + A056490(n). (End)
%e A056344 For a(4)=3, the arrangements are ABCD, ABDC, and ACBD, all chiral, their reverses being ADCB, ACDB, and ADBC respectively.
%t A056344 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 A056344 T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
%t A056344 a[n_] := T[n, 4];
%t A056344 Array[a, 24] (* _Jean-François Alcover_, Nov 05 2017, after _Andrew Howroyd_ *)
%t A056344 k=4; 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 A056344 Column 4 of A273891.
%Y A056344 Cf. A056284 (oriented), A056490 (achiral), A305543 (chiral).
%K A056344 nonn
%O A056344 1,4
%A A056344 _Marks R. Nester_