cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Robert A. Russell, Oct 18 2018

Keywords

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.

Examples

			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.

Links

Crossrefs

Columns 1-6 are A000004, A059053, A320643, A320644, A320645, A320646.
Row sums are A320749.
Cf. A152175 (oriented), A152176 (unoriented), A304972 (achiral).

Programs

  • 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
  • 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

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)).

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).

Original entry on oeis.org

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
Offset: 1

Views

Author

Robert A. Russell, Oct 21 2018

Keywords

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.

Examples

			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.

Links

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).

Programs

  • 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
  • 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

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).

A327740 Number of n-bead bracelet structures using an infinite alphabet that are not self-equivalent under either a nonzero rotation or reversal (turning over bracelet).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 185, 1011, 5304, 29740, 172037, 1055232, 6790613, 46034820
Offset: 1

Views

Author

Andrew Howroyd, Sep 23 2019

Keywords

Crossrefs

Row sums of A324802.
Cf. A320749, A327696.

Formula

a(p) = A320749(p) for prime p.
Showing 1-3 of 3 results.

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