A283877 Number of non-isomorphic set-systems of weight n.

1, 1, 2, 4, 9, 18, 44, 98, 244, 605, 1595, 4273, 12048, 34790, 104480, 322954, 1031556, 3389413, 11464454, 39820812, 141962355, 518663683, 1940341269, 7424565391, 29033121685, 115921101414, 472219204088, 1961177127371, 8298334192288, 35751364047676, 156736154469354

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

Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.

Sequence in context: A000678 A362037 A368422 * A331850 A081490 A292478

Adjacent sequences: A283874 A283875 A283876 * A283878 A283879 A283880

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