OFFSET
0,1
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..444
Equational Theories project, Basic theory of magmas.
Equational Theories project, Generating a list of equations on magmas, Python script.
Terence Tao, A pilot project in universal algebra to explore new ways to collaborate and use machine assistance?, 25 Sep 2024.
FORMULA
EXAMPLE
For n=0 the distinct laws are x=x and x=y.
For n=1 the distinct laws are x=x*x, x=x*y, x=y*x, x=y*y, and x=y*z.
For n=2 there are 39 distinct laws, including for instance x=(x*y)*x, x*y=y*z, and x*y=y*x, but not x*y=x*y because this is a nontrivial reflexive law (of the form X=X for an expression X that is not just a single indeterminate).
PROG
(Python)
from functools import lru_cache
from sympy.functions.combinatorial.numbers import stirling, bell, catalan
def A376640(n):
if n&1:
return catalan(n+1)*bell(n+2)>>1
else:
if not n: return 2
@lru_cache(maxsize=None)
def ach(n, k): return (n==k) if n<2 else k*ach(n-2, k)+ach(n-2, k-1)+ach(n-2, k-2)
return (catalan(n+1)*bell(n+2)+catalan(m:=n>>1)*((sum(stirling(n+2, k, kind=2)+ach(n+2, k)>>1 for k in range(n+3))<<1)-bell(n+2)-(bell(m+1)<<1))>>1) # Chai Wah Wu, Oct 15 2024
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Terence Tao, Sep 30 2024
EXTENSIONS
a(7) and beyond from Michael S. Branicky, Sep 30 2024
STATUS
approved