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 1-5 of 5 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)).

A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 13, 46, 144, 420, 1221, 3474, 9856, 27794, 78632, 222156, 629760, 1787440, 5087797, 14509580, 41479867, 118811286, 341009901, 980488510, 2824029648, 8146494860, 23534997912, 68084154502, 197211336576, 571915188840, 1660405181149, 4825559508106, 14038010213051, 40875403561680, 119122661856133, 347441159864556, 1014152747485696
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.
There are nonrecursive formulas, generating functions, and computer programs for A002076 and A182522, which can be used in conjunction with the first formula.

Examples

			For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.

Links

Crossrefs

Column 3 of A320742.
Cf. A002076 (oriented), A056353 (unoriented), A182522 (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 *)
k=3; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

Formula

a(n) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, 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)).
a(n) = A059053(n) + A320643(n).

A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 30, 130, 532, 2006, 7626, 28401, 106260, 396435, 1486147, 5580130, 21032880, 79486763, 301317282, 1145123672, 4362804633, 16658456825, 63738451998, 244332656201, 938244497740, 3608640426930, 13899977105315, 53614228550220, 207061964668740, 800639722002163, 3099251007215286, 12009598156277090, 46582685655751645, 180850428684482360
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.
There are nonrecursive formulas, generating functions, and computer programs for A056292 and A305750, which can be used in conjunction with the first formula.

Examples

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Links

Crossrefs

Column 4 of A320742.
Cf. A056292 (oriented), A056354 (unoriented), A305750 (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 *)
k=4; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

Formula

a(n) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=4 is the maximum number of colors, 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)).
a(n) = A059053(n) + A320643(n) + A320644(n).

A320745 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 181, 871, 4016, 18526, 85101, 393148, 1822977, 8500893, 39809180, 187230704, 883730048, 4184926222, 19874478310, 94629276256, 451604739323, 2159748985582, 10348493650194, 49671898709098, 238804606717950, 1149792470325340, 5543620159707666, 26762240285558924, 129350640352555296, 625889650880647630, 3031651402693863747, 14698911258326292182, 71332938143655936584, 346474231506471943759
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.
There are nonrecursive formulas, generating functions, and computer programs for A056293 and A305751, which can be used in conjunction with the first formula.

Examples

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Links

Crossrefs

Column 5 of A320742.
Cf. A056293 (oriented), A056355 (unoriented), A305751 (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 *)
k=5; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

Formula

a(n) = (A056293(n) - A305751(n)) / 2 = A056293(n) - A056355(n) = A056355(n) - A305751(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=5 is the maximum number of colors, 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)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n).

A320746 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 34, 190, 996, 5070, 26454, 139484, 749742, 4082481, 22509626, 125231540, 702004040, 3958071545, 22423227634, 127524417922, 727617119592, 4163076477731, 23876455868772, 137228326265794, 790200053665362, 4557942281943078, 26331297198477874, 152331940294133402, 882422871962784662, 5117852332008063806, 29715786649820358328, 172720006045619486686, 1004904748993330281274, 5852047136464153694312
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.
There are nonrecursive formulas, generating functions, and computer programs for A056294 and A305752, which can be used in conjunction with the first formula.

Examples

			For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC.

Links

Crossrefs

Column 6 of A320742.
Cf. A056294 (oriented), A056356 (unoriented), A305752 (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 *)
k=6; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

Formula

a(n) = (A056294(n) - A305752(n)) / 2 = A056294(n) - A056356(n) = A056356(n) - A305752(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=6 is the maximum number of colors, 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)).
a(n) = A059053(n) + A320643(n) + A320644(n) + A320645(n) + A320646(n).
Showing 1-5 of 5 results.

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