OFFSET
0,3
COMMENTS
A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
FORMULA
Euler transform of A300913.
EXAMPLE
Non-isomorphic representatives of the a(4)=9 set-systems are:
((1234)),
((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
((1)(2)(3)(4)).
PROG
(PARI)
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t), x, x^t) )); s+=permcount(q)*polcoef(exp(g - subst(g, x, x^2)), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 17 2017
EXTENSIONS
a(0) = 1 prepended and terms a(11) and beyond from Andrew Howroyd, Sep 01 2019
STATUS
approved