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.

A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

1, 6, 996, 31857648, 266666713602640, 929809173755713574913480, 2002123402266181527640478418179038176, 3702236248557739850415303240942330019881771301360640, 7805296829528400289943264314587254996361382902046539931447903763389056
Offset: 1

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Author

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

Keywords

Comments

A079187(n)+A079190(n)=A001329(n).
Each a(n) is equal to the sum of the elements in row n of A079191.

Links

Crossrefs

Cf. A079187, A079189, A079191.

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = Product_{i>=1, j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (Sum_{d|i} (d*s_d))^(s_i*(i*s_i+1)/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (Sum_{d|i and i/d is odd} (d*s_d))^s_i * (Sum_{d|i} (d*s_d))^(i*s_i^2/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). [Corrected by Sean A. Irvine, Aug 03 2025]
a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)

Extensions

Edited, corrected and extended with formula by Christian G. Bower, Dec 12 2003
a(9) from Sean A. Irvine, Aug 03 2025

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

A079188 Number of isomorphism classes of non-anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 0, 4, 1, 4, 44, 2285, 0, 0, 0, 24, 64, 212, 35240, 147088764
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!)
A079176(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 of this sequence is given by A079177(n).

Examples

			First four rows:
  0;
  0, 4;
  1, 4, 44, 2285;
  0, 0, 0, 24, 64, 212, 35240, 147088764.

Links

Crossrefs

Cf. A027423, A079171, A079186, A079187, A079191.

Formula

a(n) = A079171(n) - A079191(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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