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 1-4 of 4 results.

A076113 a(n) = n^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 2, 27, 4096, 9765625, 470184984576, 558545864083284007, 19342813113834066795298816, 22528399544939174411840147874772641, 1000000000000000000000000000000000000000000000, 1890591424712781041871514584574319778449301246603238034051
Offset: 0

Views

Author

Amarnath Murthy, Oct 09 2002

Keywords

Comments

Number of labeled commutative idempotent groupoids with n elements. [edited by Michel Marcus, Jul 10 2025]
Product of terms in n-th row of A076112.

Links

Crossrefs

Cf. A000169, A002489, A006125, A023037, A030257, A076112, A090588.

Programs

  • PARI
    a(n) = n^(n*(n-1)/2); \\ Joerg Arndt, Nov 04 2013

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
a(0)=1 prepended by Alois P. Heinz, Jun 30 2022

A038021 Triangle: T(n,k), k<=n: commutative groupoids with n elements and k idempotents.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 38, 57, 27, 7, 13872, 18544, 9280, 2080, 192, 83360520, 104208775, 52110500, 13035000, 1632750, 82355
Offset: 0

Views

Author

Christian G. Bower, May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A001425, A030257, A030260, A030263.

A038017 Number of n-element commutative groupoids with an identity ("pointed" groupoids).

Original entry on oeis.org

1, 2, 15, 720, 409600, 3920030472, 775775333825891, 3837862827737186253664, 558740081065710564284870598075, 2755731923933734753149997221152548428020, 520996314135332606285488148844494695722050333912483
Offset: 1

Views

Author

Christian G. Bower, May 15 1998; revised Dec 05 2003

Keywords

Comments

Also number of commutative partial groupoids with n-1 elements or commutative groupoids with an absorbant (zero) element with n elements.

Links

Crossrefs

Cf. A001329, A030257.

Formula

a(n+1) = 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, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (1 + sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (1 + sum {d|i} (d*s_d))^(i*s_i^2/2) * (1 + sum {d|i/2} (d*s_d))^s_i or {i != j} (1 + sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
a(n) asymptotic to (n^binomial(n, 2)+1)/n! = A090599(n)/A000142(n) = A076113(n)/A000142(n-1)

A030259 Number of nonisomorphic commutative idempotent groupoids with a nontrivial symmetry.

Original entry on oeis.org

0, 0, 0, 4, 38, 1894
Offset: 0

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

A030257 - A030258.
Showing 1-4 of 4 results.

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

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