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A320742 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). 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,34

COMMENTS

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.

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) = 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).

EXAMPLE

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.

MATHEMATICA

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

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)={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

CROSSREFS

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

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, Oct 21 2018

STATUS

approved

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Last modified July 24 11:17 EDT 2021. Contains 346273 sequences. (Running on oeis4.)