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%I #16 Jul 12 2018 08:55:17
%S 1,0,1,2,5,11,30,78,223,658,2026,6429,21015,70233,239360,829224,
%T 2912947,10356334,37205121,134887153,493000086,1814902409,6724595543,
%U 25061885217,93899071368,353514105817,1336822098961,5075833932200
%N A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.
%C a(n) is the number of hyperforests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A134955(n-1) counts the hyperforests of order n with one or more isolated nodes.
%F Euler transform of b(1) = 0, b(n > 1) = A035053(n). - _Gus Wiseman_, May 21 2018
%e From _Gus Wiseman_, May 21 2018: (Start)
%e Non-isomorphic representatives of the a(5) = 11 hyperforests are the following:
%e {{1,2,3,4,5}}
%e {{1,2},{3,4,5}}
%e {{1,5},{2,3,4,5}}
%e {{1,2,5},{3,4,5}}
%e {{1,2},{2,5},{3,4,5}}
%e {{1,2},{3,5},{4,5}}
%e {{1,4},{2,5},{3,4,5}}
%e {{1,5},{2,5},{3,4,5}}
%e {{1,3},{2,4},{3,5},{4,5}}
%e {{1,4},{2,5},{3,5},{4,5}}
%e {{1,5},{2,5},{3,5},{4,5}}
%e (End)
%t etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b];
%t b[0] = 0; b[n_] := b[n] = etr[etr[b]][n-1];
%t c[1] = 0; c[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}];
%t a = etr[c];
%t Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jul 12 2018, after _Alois P. Heinz_'s code for A035053 *)
%o (PARI) \\ here b is A007563 as vector
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o b(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
%o seq(n)={my(u=b(n)); concat([1], EulerT(concat([0], Vec(Ser(EulerT(u))*(1-x*Ser(u))-1))))} \\ _Andrew Howroyd_, May 22 2018
%Y Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144958 (unlabeled forests without isolated vertices), A144959, A304716, A304717, A304867, A304911.
%K nonn
%O 0,4
%A _Washington Bomfim_, Sep 27 2008
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