OFFSET
2,6
COMMENTS
P. Freyd (see link) writes: "Bower’s sequence A050365 can be interpreted to count the number of anti-symmetric elhsls (and elhas), ... those for which all automorphisms are one." Here 'elha' stands for 'equationally linear Heyting algebra' and 'elhsl' means 'equationally linear Heyting semi-lattice'. Then Freyd gives another interpretation and coins the name Bower-rank. "For a set of finite rank we may define its Bower-rank as the first ordinal larger than the product of the Bower-ranks of its elements. Then A050365 counts the number of sets of given Bower-rank." - Peter Luschny, Nov 13 2018
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..10000
Peter Freyd, On the size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras, July 17 2017.
FORMULA
Shifts left under transform T where Ta has Dirichlet g.f. Product_{n>=1}(1+1/n^s)^a(n).
EXAMPLE
The different ways of writing the numbers 2 through 7 as identity mterms are:
2 = 2,
3 = 1 + 2,
4 = 1 + (1+2),
5 = 1 + (1+1+2),
6 = 1 + (1+1+1+2),
7 = 1 + (1+1+1+1+2) = 1 + 2*(1+2).
PROG
(PARI) seq(n)={my(v=vector(n, i, i==1)); for(k=2, n, v=dirmul(v, vector(#v, i, my(e=valuation(i, k)); if(i==k^e, binomial(v[k-1], e), 0)))); v} \\ Andrew Howroyd, Nov 17 2018
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Christian G. Bower, Oct 15 1999
STATUS
approved