cp's OEIS Frontend

A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

%I A320743 #15 Nov 04 2019 12:45:42
%S A320743 0,0,0,0,0,4,13,46,144,420,1221,3474,9856,27794,78632,222156,629760,
%T A320743 1787440,5087797,14509580,41479867,118811286,341009901,980488510,
%U A320743 2824029648,8146494860,23534997912,68084154502,197211336576,571915188840,1660405181149,4825559508106,14038010213051,40875403561680,119122661856133,347441159864556,1014152747485696
%N A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).
%C A320743 Two color patterns are equivalent if the colors are permuted.
%C A320743 Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%C A320743 There are nonrecursive formulas, generating functions, and computer programs for A002076 and A182522, which can be used in conjunction with the first formula.
%H A320743 Andrew Howroyd, <a href="/A320743/b320743.txt">Table of n, a(n) for n = 1..200</a>
%H A320743 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F A320743 a(n) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n).
%F A320743 a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).
%F A320743 a(n) = A059053(n) + A320643(n).
%e A320743 For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
%t A320743 Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#]&], Boole[n == 0 && k == 0]]
%t A320743 Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
%t A320743 k=3; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]
%Y A320743 Column 3 of A320742.
%Y A320743 Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral).
%K A320743 nonn,easy
%O A320743 1,6
%A A320743 _Robert A. Russell_, Oct 21 2018