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

Author

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

Keywords

• nonn

Comments

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

References

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

Links

• Index entries for sequences related to semigroups

Crossrefs

Cf. A005345, A030449, A258621.

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

Programs

Mathematica

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 *)

PARI

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

Sage

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

Formula

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