A005345 -id:A005345 - cp's OEIS frontend

A030450 Related to number of elements in the free band (idempotent semigroup) on n generators.

1, 1, 4, 144, 331776, 2751882854400, 272622932796264897576960000, 3641839910835401567626683591527643364677019238400000000

Offset: 0

Marcel Jackson (marcel_j(AT)hilbert.maths.utas.edu.au)

nonn

Continued square root 2 = sqrt(1 + sqrt(1 + sqrt(4 + sqrt(144 + ...)))) = sqrt(1 + sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + ...))))) [S. Ramanujan]. - Michael Somos, Dec 03 2017

John M. Howie, Fundamentals of Semigroup Theory, Oxford University Press 1995, p. 123.

Index entries for sequences related to semigroups

Cf. A005345, A030449, A258621.

A052129(n) = n*a(n-1) if n > 0.

s=1;lst={};Do[AppendTo[lst,s*=s*=n],{n,9}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 20 2009 *)
Fold[Append[#1, (#2 Last[#1])^2] &, {1}, Range@ 7] (* Michael De Vlieger, Dec 03 2017 *)

{a(n) = if(n<0, 0, prod(i=1, n, (n-i+1)^2^i))}; /* Michael Somos, Oct 22 2006 */

def A030450(n) :
   return prod((n-i+1)^(2^i) for i in (1..n))
[A030450(n) for n in (0..9)] # Jani Melik, Jun 06 2015

Binomial transform is A005345. - Michael Somos, Oct 22 2006
a(n) = (n*a(n-1))^2 if n > 0. a(0)=1. - Michael Somos, Oct 22 2006
a(n) = Product_{i=1..n} (n-i+1)^(2^i).
Sum_{n>=1} 1/a(n) = A258621. - Amiram Eldar, Nov 19 2020

A030449 Number of elements in the free band (idempotent semigroup) on n generators.

Original entry on oeis.org

1, 6, 159, 332380, 2751884514765, 272622932796281408879065986, 3641839910835401567626683593436003894250931310990279691

Offset: 1

marcel_j(AT)hilbert.maths.utas.edu.au (Marcel Jackson)

nonn

An idempotent semigroup satisfies the equation xx=x for any element x.

J. Howie, Fundamentals of Semigroup Theory, Oxford University Press 1995, p. 123.

Index entries for sequences related to semigroups

Cf. A030450. A005345(n)=a(n)+1.

a(n) = Sum_{k=1..n} C(n, k) A030450(k).

A376025 Number of elements of the free multiplicatively idempotent rig on n generators.

Original entry on oeis.org

4, 13, 284, 510605

Offset: 0

Morgan Rogers, Sep 06 2024

Let T_n denote the free idempotent monoid on n generators and alpha: T_n -> P([n]) the function sending an element to the generators appearing in it. Then this sequence is computed as the number of triples (S,D,p), where S is a "replete" subsemigroup of T_n (i.e. a subsemigroup for which (xuy) and (xvy) in S implies (xuvy) is in S), D is a "sparse" subset of T_n (i.e., for s and t in D, alpha(s) a subset of alpha(t) implies s = t), S "dominates" D (i.e. for s in S and t in D, alpha(s) is not a subset of alpha(t) and S is closed under multiplication by elements of D) and p is a "parity" function from the image of S under alpha to {0,1}. See Rogers 2024.

			For n = 0, the free idempotent rig on zero generators is the quotient of the natural numbers by the congruence generated by x ~ x^2. Considering different values of x, this yields the trivial relations 0 ~ 0 and 1 ~ 1, then 2 ~ 4, whence x+2 ~ x+4 for every x. By a parity argument and induction, this entirely determines the congruence: 2 is related to every larger even number and 3 is related to every larger odd number. Thus the resulting rig thus has 4 elements: 0, 1 and the equivalence classes of 2 and 3.

Morgan Rogers, From free idempotent monoids to free multiplicatively idempotent rigs, arXiv:2408.17440 [math.RA], 2024, pages 28-30.

Cf. A005345.

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A030450 Related to number of elements in the free band (idempotent semigroup) on n generators.

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A030449 Number of elements in the free band (idempotent semigroup) on n generators.

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A376025 Number of elements of the free multiplicatively idempotent rig on n generators.

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