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A320742
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Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a cycle of length n using k or fewer colors (subsets).
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9
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 6, 13, 2, 0, 0, 0, 0, 0, 0, 6, 30, 46, 7, 0, 0, 0, 0, 0, 0, 6, 34, 130, 144, 12, 0, 0, 0, 0, 0, 0, 6, 34, 181, 532, 420, 31, 0, 0, 0, 0, 0, 0, 6, 34, 190, 871, 2006, 1221, 58, 0, 0, 0, 0, 0, 0, 6, 34, 190, 996, 4016, 7626, 3474, 126, 0, 0, 0, 0, 0, 0, 6, 34, 190, 1011, 5070, 18526, 28401, 9856, 234, 0
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OFFSET
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1,34
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COMMENTS
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Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
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LINKS
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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.
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FORMULA
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T(n,k) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), 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)).
T(n,k) = (A320747(n,k) - A305749(n,k)) / 2 = A320747(n,k) - A320748(n,k)= A320748(n,k) - A305749(n,k).
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EXAMPLE
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Array begins with T(1,1):
0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 ...
0 0 4 6 6 6 6 6 6 6 6 6 ...
0 1 13 30 34 34 34 34 34 34 34 34 ...
0 2 46 130 181 190 190 190 190 190 190 190 ...
0 7 144 532 871 996 1011 1011 1011 1011 1011 1011 ...
0 12 420 2006 4016 5070 5328 5352 5352 5352 5352 5352 ...
0 31 1221 7626 18526 26454 29215 29705 29740 29740 29740 29740 ...
0 58 3474 28401 85101 139484 165164 171556 172415 172466 172466 172466 ...
For T(6,4)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD and AABCBD-AABCDC.
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MATHEMATICA
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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]]
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 *)
Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k-n+1}], {k, 15}, {n, k}] // Flatten
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PROG
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(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)={my(M=(R(n) - Ach(n))/2); for(i=2, n, M[, i] += M[, i-1]); M}
{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019
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CROSSREFS
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Partial row sums of A320647.
Columns 1-6 are A000004, A059053, A320743, A320744, A320745, A320746
For increasing k, columns converge to A320749.
Cf. A320747 (oriented), A320748 (unoriented), A305749 (achiral).
Sequence in context: A051390 A124120 A324803 * A093318 A255329 A127560
Adjacent sequences: A320739 A320740 A320741 * A320743 A320744 A320745
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Robert A. Russell, Oct 21 2018
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STATUS
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approved
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