login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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