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 1-10 of 40 results. Next

A079171 Number of isomorphism classes of closed binary operations (groupoids) on a set of order n, listed by class size.

Original entry on oeis.org

1, 4, 6, 3, 12, 78, 3237, 2, 1, 14, 30, 275, 495, 48810, 178932325
Offset: 1

Views

Author

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

Keywords

Comments

Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
A002489(n) is equal to the sum of the products of each element in row n of this sequence and the corresponding element of A079210.
The sum of each row n is given by A001329(n).

Examples

			First four rows:
  1;
  4, 6;
  3, 12, 78, 3237;
  2, 1, 14, 30, 275, 495, 48810, 178932325.

Links

Crossrefs

Cf. A002489, A001329. a(n, A027423(n)) = A030245(n).

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Original entry on oeis.org

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448
Offset: 0

Views

Author

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Keywords

Links

Crossrefs

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.
Cf. A002489, A079192, A079195, A079198, A079201, A079210.

Formula

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).
a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

Extensions

a(8) from Andrew Howroyd, Jan 26 2022
a(9) from Andrew Howroyd, Feb 14 2022

A079195 Number of non-associative commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 666, 1047436, 30517547395, 21936950639192784, 459986536544739894613595, 324518553658426726783148869733112
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Cf. A023813, A023815, A079192, A079196 (isomorphism classes), A079197, A079198.

Formula

A079192(n) + a(n) + A079198(n) + A023815(n) = A002489(n).
a(n) = Sum_{k>=1} A079197(n,k)*A079210(n,k).
a(n) = A023813(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

A079197 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 1, 1, 4, 5, 107, 0, 0, 0, 5, 0, 28, 488, 43389
Offset: 1

Views

Author

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

Keywords

Comments

A079194(n)+A079197(n)+A079200(n)+A079201(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; 0,1; 1,4,5,107; 0,0,0,5,0,28,488,43389
A079195(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 A079196(x).

Links

Crossrefs

Cf. A079194, A079198, A079199, A079200, A079201.

A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 6, 24, 1, 2, 6, 24, 120, 1, 2, 6, 24, 120, 720, 1, 2, 6, 24, 120, 720, 5040, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 1, 2, 6, 24, 120, 720, 5040, 40320
Offset: 1

Views

Author

Paul Curtz, Oct 12 2009

Keywords

Examples

			Triangle begins:
  1;
  1, 2;
  1, 2, 6;
  1, 2, 6, 24;
  1, 2, 6, 24, 120;
  1, 2, 6, 24, 120, 720;
  1, 2, 6, 24, 120, 720, 5040;
  ...

Links

Crossrefs

Cf. A000142, A002260, A094310, A077012, A079210.
Cf. A014454.
Row sums give A007489.

Programs

  • Haskell
    import Data.List (inits)
a166350 n k = a166350_tabl !! (n-1) !! (n-1)
a166350_row n = a166350_tabl !! (n-1)
a166350_tabl = tail $ inits $ tail a000142_list
-- Reinhard Zumkeller, Nov 11 2013
  • Mathematica
    Flatten[Table[Range[n]!,{n,11}]] (* Harvey P. Dale, Jan 06 2012 *)
Module[{nn=20,fs},fs=Range[nn]!;Table[Take[fs,n],{n,nn}]]//Flatten (* Harvey P. Dale, Jun 14 2020 *)

Formula

T(n,m) = A000142(m).

Extensions

Definition clarified - R. J. Mathar, Oct 14 2009

A079202 Number of isomorphism classes of non-associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 32, 2155, 0, 0, 0, 12, 60, 184, 34544, 147032271
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(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; 0,0; 0,0,32,2155; 0,0,0,12,60,184,34544,147032271
A079230(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 A079231(x).

Links

Crossrefs

Cf. A079203, A079204, A079205, A079197, A079207, A079208, A079209, A079230, A079231.

A079203 Number of isomorphism classes of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 2, 0, 6, 34, 952, 0, 1, 12, 6, 181, 283, 13333, 31839187
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(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; 0,2; 0,6,34,952; 0,1,12,6,181,283,13333,31839187
A079232(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 A079233(x).

Links

Crossrefs

Cf. A079202, A079204, A079205, A079197, A079207, A079208, A079209, A079232, A079233.

A079204 Number of isomorphism classes of non-associative non-commutative anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 146, 12992
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(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; 0,0; 0,0,0,8; 0,0,0,0,0,0,146,12992
A079234(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 A079235(x).

Links

Crossrefs

Cf. A079202, A079203, A079205, A079197, A079207, A079208, A079209, A079234, A079235.

A079205 Number of isomorphism classes of non-associative non-commutative anti-associative anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 29, 0, 237, 4374
Offset: 1

Views

Author

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

Keywords

Comments

A079202(n)+A079203(n)+A079204(n)+A079205(n)+A079197(n)+A079207(n)+A079208(n)+A079209(n)+A063524(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; 2,0; 0,2,0,0; 0,0,2,0,29,0,237,4374
A079236(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 A079237(x).

Links

Crossrefs

Cf. A079202, A079203, A079204, A079197, A079207, A079208, A079209, A079236, A079237.

A079207 Number of isomorphism classes of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, 4, 4, 0, 46, 73, 0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 36, 0, 0, 0, 0, 84, 0, 0, 38, 415, 0, 758, 32, 6682, 18426, 0, 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

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;
  0, 0;
  0, 0, 4, 6;
  0, 0, 0, 4, 4, 0, 46, 73;
  0, 0, 0, 0, 4, 0, 0, 8, 0, 2, 36, 0, 43, 2, 473, 1020;
  ...

Links

Crossrefs

Row sums give A079241.
Cf. A027423 (row lengths), A079175, A079201, A079202, A079203, A079204, A079205, A079197, A079208, A079209, A079240.

Formula

A079202(n,k) + A079203(n,k) + A079204(n,k) + A079205(n,k) + A079197(n,k) + A079208(n,k) + T(n,k) + A079201(n,k) = A079171(n,k).
A079240(n) = Sum_{k>=1} T(n,k)*A079210(n,k).
T(n,k) = A079175(n,k) - A079201(n,k) - A079208(n,k). - Andrew Howroyd, Jan 27 2022

Extensions

a(0)=0 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 27 2022
Showing 1-10 of 40 results. Next

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