cp's OEIS Frontend

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

%I A283877 #29 Jan 16 2024 17:42:29
%S A283877 1,1,2,4,9,18,44,98,244,605,1595,4273,12048,34790,104480,322954,
%T A283877 1031556,3389413,11464454,39820812,141962355,518663683,1940341269,
%U A283877 7424565391,29033121685,115921101414,472219204088,1961177127371,8298334192288,35751364047676,156736154469354
%N A283877 Number of non-isomorphic set-systems of weight n.
%C A283877 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.
%H A283877 Andrew Howroyd, <a href="/A283877/b283877.txt">Table of n, a(n) for n = 0..50</a>
%F A283877 Euler transform of A300913.
%e A283877 Non-isomorphic representatives of the a(4)=9 set-systems are:
%e A283877 ((1234)),
%e A283877 ((1)(234)), ((3)(123)), ((12)(34)), ((13)(23)),
%e A283877 ((1)(2)(12)), ((1)(2)(34)), ((1)(3)(23)),
%e A283877 ((1)(2)(3)(4)).
%o A283877 (PARI)
%o A283877 WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o A283877 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}
%o A283877 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))}
%o A283877 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
%Y A283877 Cf. A007716, A034691, A049311, A056156, A089259, A116540, A300913.
%K A283877 nonn
%O A283877 0,3
%A A283877 _Gus Wiseman_, Mar 17 2017
%E A283877 a(0) = 1 prepended and terms a(11) and beyond from _Andrew Howroyd_, Sep 01 2019