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%I #21 Nov 04 2019 02:17:30
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,2,0,0,0,1,12,17,4,0,0,0,2,44,84,
%T 51,9,0,0,0,7,137,388,339,125,15,0,0,0,12,408,1586,2010,1054,258,24,0,
%U 0,0,31,1190,6405,10900,7928,2761,490,35,0,0,0,58,3416,24927,56700,54383,25680,6392,859,51,0,0,0,126,9730,96404,286888,356594,218246,72284,13472,1420,68,0,0
%N Triangle read by rows: T(n,k) is the number of chiral pairs of cycles of length n (1) with a color pattern of exactly k colors or equivalently (2) partitioned into k nonempty subsets.
%C Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
%C Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
%H Andrew Howroyd, <a href="/A320647/b320647.txt">Table of n, a(n) for n = 1..1275</a>
%H 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 T(n,k) = (A152175(n,k) - A304972(n,k)) / 2 = A152175(n,k) - A152176(n,k) = A152176(n,k) - A304972(n,k).
%F T(n,k) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where 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)).
%e The triangle begins with T(1,1):
%e 0;
%e 0, 0;
%e 0, 0, 0;
%e 0, 0, 0, 0;
%e 0, 0, 0, 0, 0;
%e 0, 0, 4, 2, 0, 0;
%e 0, 1, 12, 17, 4, 0, 0;
%e 0, 2, 44, 84, 51, 9, 0, 0;
%e 0, 7, 137, 388, 339, 125, 15, 0, 0;
%e 0, 12, 408, 1586, 2010, 1054, 258, 24, 0, 0;
%e 0, 31, 1190, 6405, 10900, 7928, 2761, 490, 35, 0, 0;
%e 0, 58, 3416, 24927, 56700, 54383, 25680, 6392, 859, 51, 0, 0;
%e 0, 126, 9730, 96404, 286888, 356594, 218246, 72284, 13472, 1420, 68, 0, 0;
%e ...
%e For T(6,3)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.
%e For T(6,4)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC.
%t 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 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 Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n)-Ach[n,k]/2,{n,12},{k,n}] // Flatten
%o (PARI) \\ Ach is A304972 and R is A152175 as square matrices.
%o Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
%o R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
%o T(n)={(R(n) - Ach(n))/2}
%o { my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Sep 20 2019
%Y Columns 1-6 are A000004, A059053, A320643, A320644, A320645, A320646.
%Y Row sums are A320749.
%Y Cf. A152175 (oriented), A152176 (unoriented), A304972 (achiral).
%K nonn,tabl,easy
%O 1,18
%A _Robert A. Russell_, Oct 18 2018
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