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 1-7 of 7 results.

A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 8, 18, 12, 1, 18, 78, 120, 60, 1, 34, 273, 780, 900, 360, 1, 70, 921, 4212, 8400, 7560, 2520, 1, 134, 2916, 20424, 63000, 95760, 70560, 20160, 1, 270, 9150, 93360, 417120, 952560, 1164240, 725760, 181440, 1, 526, 28065, 409380, 2551560, 8217720, 14817600, 15120000, 8164800, 1814400, 1, 1054, 85773, 1749780, 14804700, 64615680, 161247240, 239500800, 209563200, 99792000, 19958400
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Examples

			The triangle begins:
  1;
  1,   1;
  1,   4,    3;
  1,   8,   18,      12;
  1,  18,   78,     120,      60;
  1,  34,  273,     780,     900,     360;
  1,  70,  921,    4212,    8400,    7560,     2520;
  1, 134, 2916,   20424,   63000,   95760,    70560,    20160;
  1, 270, 9150,   93360,  417120,  952560,  1164240,   725760,  181440;
  ...
For T(3,2)=4, the achiral color rows are ABA and BAB, while the chiral pairs are AAB-BAA and ABB-BBA. For T(3,3)=3, the color rows are all chiral pairs: ABC-CBA, ACB-BCA, and BAC-CAB.

Links

Crossrefs

Columns 1-6 are A057427, A056309, A056310, A056311, A056312, and A056313.
Row sums are A326963.
A019538 counts chiral pairs as two, i.e., the rows are not reversible.
Cf. A277504, A305622.

Programs

  • Mathematica
    Table[(k!/2) (StirlingS2[n, k] + StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten
  • PARI
    T(n,k) = {k! * (stirling(n,k,2) + stirling((n+1)\2,k,2)) / 2} \\ Andrew Howroyd, Sep 13 2019

Formula

T(n,k) = (k!/2) * (S2(n,k) + S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
T(n,k) = (A019538(n,k) + A019538(ceiling(n/2),k)) / 2.
G.f. for column k: k! x^k / (2*Product_{i=1..k}(1-ix)) + k! (x^(2k-1)+x^(2k)) / (2*Product{i=1..k}(1-i x^2)). - Robert A. Russell, Sep 25 2018
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A277504(n, i). - Andrew Howroyd, Sep 13 2019

A305623 Number of chiral pairs of rows of n colors with exactly 3 different colors.

Original entry on oeis.org

0, 0, 3, 18, 72, 267, 885, 2880, 9000, 27915, 85233, 259308, 783972, 2366007, 7122405, 21422160, 64364400, 193307955, 580316313, 1741791348, 5226945372, 15684152847, 47058746925, 141189342840, 423593188200, 1270831465995, 3812595048993, 11437991207388, 34314376250772, 102943948309287, 308833455491445, 926503630549920, 2779517334002400, 8338565015656035, 25015720816575273, 75047214375967428
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			For a(3) = 3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.

Links

Crossrefs

Third column of A305622.
A056454(n) is number of achiral rows of n colors with exactly 3 different colors.
Cf. A001117, A056310.

Programs

  • Mathematica
    k=3; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
  • PARI
    a(n) = 3*(stirling(n,3,2)-stirling(ceil(n/2),3,2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=3 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A001117(n) - A056454(n)) / 2.
a(n) = A001117(n) - A056310(n) = A056310(n) - A056454(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=3 colors used.
G.f.: 3*x^3*(5*x^2-x-1)/(-36*x^6+30*x^5+24*x^4-25*x^3-x^2+5*x-1). - Simon Plouffe, Jun 20 2018

A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

Original entry on oeis.org

0, 0, 0, 12, 120, 780, 4188, 20400, 93120, 409140, 1748220, 7337232, 30386160, 124696740, 508250988, 2061566400, 8331954240, 33585590580, 135115594140, 542784981552, 2178107091600, 8733341736900, 34996103558988, 140172672276000, 561255446475360, 2246716252964820, 8991948337723260, 35983044114659472, 143977928423467440, 576048972752188260, 2304607666801990188, 9219666007300387200, 36882370043723748480
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			For a(4) = 12, the chiral pairs are the 4! = 24 permutations of ABCD, each paired with its reverse.

Crossrefs

Fourth column of A305622.
A056455(n) is number of achiral rows of n colors with exactly 4 different colors.
Cf. A000919, A056311.

Programs

  • Mathematica
    k=4; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
  • PARI
    a(n) = my(k=4); (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2)); \\ Michel Marcus, Jun 07 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=4 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000919(n) - A056455(n)) / 2.
a(n) = A000919(n) - A056311(n) = A056311(n) - A056455(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=4 colors used.

A305625 Number of chiral pairs of rows of n colors with exactly 5 different colors.

Original entry on oeis.org

0, 0, 0, 0, 60, 900, 8400, 63000, 417000, 2551440, 14802900, 82763100, 450501660, 2404493700, 12645952200, 65771370000, 339164682000, 1737485315640, 8855354531100, 44952362878500, 227475739300260, 1148269299919500, 5785013208282000, 29100046926951000, 146201097996135000, 733811769167043840, 3680292427100043300, 18446421887430345900, 92412024657725026860, 462780012983867889300, 2316780309783100387800
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			For a(5) = 60, the chiral pairs are the 5! = 120 permutations of ABCDE, each paired with its reverse.

Crossrefs

Fifth column of A305622.
A056456(n) is number of achiral rows of n colors with exactly 5 different colors.
Cf. A001118, A056312.

Programs

  • Mathematica
    k=5; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
  • PARI
    a(n) = 60*(stirling(n, 5, 2) - stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=5 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A001118(n) - A056456(n)) / 2.
a(n) = A001118(n) - A056312(n) = A056312(n) - A056456(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=5 colors used.

A305626 Number of chiral pairs of rows of n colors with exactly 6 different colors.

Original entry on oeis.org

0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64614960, 476514360, 3355664760, 22837086720, 151449482520, 984573465120, 6302069010720, 39847409421480, 249509368422720, 1550188394120520, 9570844541994120, 58789922099665680, 359629148397511080, 2192484972513916080, 13329510116645202480, 80854267307329446840, 489528474458978944080, 2959252601445086408280, 17866194139995100525080, 107751636988750077294240, 649286502010403671101240
Offset: 1

Views

Author

Robert A. Russell, Jun 06 2018

Keywords

Comments

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			For a(6) = 360, the chiral pairs are the 6! = 720 permutations of ABCDEF, each paired with its reverse.

Crossrefs

Sixth column of A305622.
A056457(n) is number of achiral rows of n colors with exactly 6 different colors.
Cf. A000920, A056313.

Programs

  • Mathematica
    k=6; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]
  • PARI
    a(n) = 360*(stirling(n, 6, 2) - stirling(ceil(n/2), 6, 2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=6 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000920(n) - A056457(n)) / 2.
a(n) = A000920(n) - A056313(n) = A056313(n) - A056457(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=6 colors used.

A327091 Number of chiral pairs of length n words with integer entries that cover an initial interval of positive integers.

Original entry on oeis.org

0, 1, 5, 36, 264, 2335, 23609, 272880, 3543360, 51123511, 811313945, 14045781456, 263429150544, 5320671461575, 115141595216009, 2657827340717760, 65185383511024320, 1692767331624879031, 46400793659613081785, 1338843898122140977776, 40562412499251225624624
Offset: 1

Views

Author

Andrew Howroyd, Sep 13 2019

Keywords

Comments

If the word is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

Examples

			a(3) = 5 because there are the following chiral pairs of words:
  112/211, 122/221,
  123/321, 132/231, 213/312.

Links

Crossrefs

Row sums of A305622.

Programs

  • PARI
    a(n) = {sum(k=1, n, k! * (stirling(n, k, 2) - stirling((n+1)\2, k, 2)) / 2)}

Formula

a(n) = Sum_{k=1..n} (k!/2) * (Stirling2(n, k) - Stirling2(ceiling(n/2), k)).

A321434 Triangle read by rows; T(n,k) is the number of achiral rows of n colors using exactly k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 6, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 126, 1806, 8400, 16800, 15120, 5040
Offset: 0

Views

Author

Robert A. Russell, Nov 09 2018

Keywords

Comments

Each zero in the data is the beginning of a new row.
Same as A131689, with rows (except for the first) repeated. - Joerg Arndt, Sep 08 2019

Examples

			The triangle begins with T(0,0):
1
0 1
0 1
0 1   2
0 1   2
0 1   6     6
0 1   6     6
0 1  14    36     24
0 1  14    36     24
0 1  30   150    240    120
0 1  30   150    240    120
0 1  62   540   1560   1800    720
0 1  62   540   1560   1800    720
0 1 126  1806   8400  16800   15120    5040
0 1 126  1806   8400  16800   15120    5040
0 1 254  5796  40824 126000  191520  141120   40320
0 1 254  5796  40824 126000  191520  141120   40320
0 1 510 18150 186480 834120 1905120 2328480 1451520 362880
For T(7,2)=14, the rows are AAABAAA, AABABAA, AABBBAA, ABAAABA, ABABABA, ABBABBA, ABBBBBA, BAAAAAB, BAABAAB, BABABAB, BABBBAB, BBAAABB, BBABABB, and BBBABBB.

Crossrefs

Columns 0-6 are A000007, A057427, A056453, A056454, A056455, A056456, A056457.
Cf. A019538 (oriented), A305621 (unoriented), A305622 (chiral).
Cf. A131689.

Programs

  • Mathematica
    Table[k! StirlingS2[Ceiling[n/2], k], {n, 0, 18}, {k, 0, (n+1)/2}] // Flatten

Formula

T(n,k) = k!*S2(ceiling(n/2),k), where S2 is the Stirling subset number A008277.
Showing 1-7 of 7 results.

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