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.

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A079242 Number of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 2, 8, 2, 122, 2, 1682
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A079208, A079243 (isomorphism classes), A079230, A079232, A079234, A079236, A079195, A079240, A079244, A063524.

Formula

A079230(n) + A079232(n) + A079234(n) + A079236(n) + A079195(n) + A079240(n) + a(n) + A079244(n) + A063524(n) = A002489(n).
a(n) = Sum_{k>=1} A079208(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended and terms a(5)-a(8) added by Andrew Howroyd, Jan 27 2022

A079198 Number of associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 50, 2352, 153002, 15876046, 7676692858
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

a(n) + A079192(n) + A079195(n) + A023815(n) = A002489(n).
Each a(n) is equal to the sum of the products of each element in row n of A079200 and the corresponding element of A079210.
Since this is the number of labeled noncommutative semigroups on an n-set, a(n) = A023814(n)-A023815(n). - Stanislav Sykora, Apr 03 2016

Links

Crossrefs

Cf. A023814, A023815, A079192, A079195, A079199, A079200.

Extensions

a(5)-a(7) added by Stanislav Sykora, Apr 03 2016

A079178 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 6, 3, 10, 78, 3229, 2, 1, 12, 30, 246, 495, 48427, 178914959
Offset: 1

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

A079178(n)+A079181(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 2,6; 3,10,78,3229; 2,1,12,30,246,495,48427,178914959
A079176(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row x of this sequence is given by A079177(x).

Links

Crossrefs

Cf. A079176, A079177, A079179.

A079192 Number of non-associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 6, 18904, 4293916368, 298023193359221998, 10314424798468598595515695154, 256923577521058877628624940679487983651948, 6277101735386680763835789098689112757675628513119817261598
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A002489, A023813, A023814, A023815, A079193 (isomorphism classes), A079194, A079195, A079198.

Formula

a(n) + A079195(n) + A079198(n) + A023815(n) = A002489(n).
a(n) = Sum_{k>=1} A079194(n,k)*A079210(n,k).
a(n) = A002489(n) - A023813(n) - A023814(n) + A023815(n). - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 26 2022

A079200 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 6, 2, 0, 0, 4, 5, 0, 46, 73, 2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 86, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Number of elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!).

Examples

			Triangle T(n,k) begins:
  0;
  0;
  2, 0;
  2, 0, 4, 6;
  2, 0, 0, 4, 5, 0, 46, 73;
  2, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Links

Crossrefs

Row sums give A079199.
Cf. A027423 (row lengths), A079171, A079194, A079175, A079197, A079198, A079199, A079201, A079210.

Formula

A079194(n,k) + A079197(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079198(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k). - Andrew Howroyd, Jan 26 2022

Extensions

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079175 Number of isomorphism classes of associative closed binary operations (semigroups) on a set of order n, listed by class size.

Original entry on oeis.org

1, 1, 2, 3, 2, 0, 7, 15, 2, 0, 0, 7, 5, 0, 62, 112, 2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221, 2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, 4, 0, 48, 0, 0, 0, 0, 92, 0, 0, 42, 506, 0, 813, 32, 7397, 19684, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

Examples

			Triangle T(n,k) begins:
  1;
  1;
  2, 3;
  2, 0, 7, 15;
  2, 0, 0, 7, 5, 0, 62, 112;
  2, 0, 0, 0, 6, 0, 0, 8, 0, 2, 51, 0, 47, 2, 576, 1221;
  ...

Links

Crossrefs

Row sums give A027851.
Cf. A023814, A027423 (row lengths), A079171, A079174, A079210.

Formula

A079174(n,k) + T(n,k) = A079171(n,k).
T(n, A027423(n)) = A058104(n).
A023814(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

Extensions

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079179 Number of anti-associative closed binary operations on a set of order n.

Original entry on oeis.org

1, 0, 2, 52, 421560
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A079176, A079180 (isomorphism classes), A079181, A079210.

Formula

a(n) = A002489(n) - A079176(n).
a(n) = Sum_{k>=1} A079181(n,k)*A079210(n,k).

Extensions

a(0)=1 prepended and a(1) corrected by Kamil Zabielski, Aug 28 2024

A079186 Number of non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 8, 13851, 3530555392, 266023223876953125, 9644962193498535546171949056, 246832875573638552740275218239438131202951, 6127827569844832702316847785612357470342156990019367075840, 193794664362053647720926884692597177807303542565053791345764052714030485961865
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A002489, A079187 (non-isomorphic), A079188, A079189, A079210.

Programs

  • PARI
    a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)) \\ Andrew Howroyd, Jan 23 2022

Formula

a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)).
a(n) = A002489(n) - A079189(n).
a(n) = Sum_{k>=1} A079178(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 23 2022

A079189 Number of anti-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

1, 1, 8, 5832, 764411904, 32000000000000000, 669462604992000000000000000, 10090701947420325348336258984797490118656, 149274165541848061518941637595308945760198454444667437056, 2832386113499265897149023834314938475799908379160975581551362823935905234944
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Links

Crossrefs

Cf. A023813, A079186, A079190 (isomorphism classes), A079191, A079210.

Programs

  • PARI
    a(n) = (n^n)*((n^2-n)^((n^2-n)/2)) \\ Andrew Howroyd, Jan 23 2022

Formula

a(n) = (n^n)*((n^2-n)^((n^2-n)/2)).
a(n) = A002489(n) - A079186(n).
a(n) = Sum_{k>=1} A079191(n,k)*A079210(n,k).
a(n) = A023813(n)*A023813(n-1).

Extensions

Edited and extended by Christian G. Bower, Dec 12 2003
a(0)=1 prepended, a(8) corrected and a(9) added by Andrew Howroyd, Jan 23 2022

A079194 Number of isomorphism classes of non-associative non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 2, 0, 8, 66, 3115, 0, 1, 14, 18, 270, 467, 48260, 178888824
Offset: 0

Views

Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

Examples

			Triangle T(n,k) begins:
  0;
  0;
  2, 2;
  0, 8, 66, 3115;
  0, 1, 14, 18, 270, 467, 48260, 178888824;
  ...

Links

Crossrefs

Row sums give A079193.
Cf. A027423 (row lengths), A079171, A079192, A079197, A079200, A079201, A079210.

Formula

T(n,k) + A079197(n,k) + A079200(n,k) + A079201(n,k) = A079171(n,k).
A079192(n,k) = Sum_{k>=1} T(n,k)*A079210(n,k).

Extensions

a(0)=0 prepended by Andrew Howroyd, Jan 26 2022
Previous Showing 21-30 of 40 results. Next

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