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 A320647 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. 10
 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, 51, 9, 0, 0, 0, 7, 137, 388, 339, 125, 15, 0, 0, 0, 12, 408, 1586, 2010, 1054, 258, 24, 0, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,18 COMMENTS Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse. Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. FORMULA T(n,k) = (A152175(n,k) - A304972(n,k)) / 2 = A152175(n,k) - A152176(n,k) = A152176(n,k) - A304972(n,k). 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)). EXAMPLE The triangle begins with T(1,1):   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,     51,      9,      0,     0;   0,   7,  137,   388,    339,    125,     15,     0,     0;   0,  12,  408,  1586,   2010,   1054,    258,    24,     0,    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;   ... For T(6,3)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC. For T(6,4)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC. MATHEMATICA 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 *) 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]] Table[DivisorSum[n, EulerPhi[#]Adnk[#, n/#, k]&]/(2n)-Ach[n, k]/2, {n, 12}, {k, n}] // Flatten PROG (PARI) \\ Ach is A304972 and R is A152175 as square matrices. 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} 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))]))} T(n)={(R(n) - Ach(n))/2} { my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019 CROSSREFS Columns 1-6 are A000004, A059053, A320643, A320644, A320645, A320646. Row sums are A320749. Cf. A152175 (oriented), A152176 (unoriented), A304972 (achiral). Sequence in context: A105087 A238012 A324802 * A028572 A107492 A159257 Adjacent sequences:  A320644 A320645 A320646 * A320648 A320649 A320650 KEYWORD nonn,tabl,easy AUTHOR Robert A. Russell, Oct 18 2018 STATUS approved

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Last modified September 27 10:28 EDT 2021. Contains 347689 sequences. (Running on oeis4.)