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A144959
A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.
19
1, 0, 1, 2, 5, 11, 30, 78, 223, 658, 2026, 6429, 21015, 70233, 239360, 829224, 2912947, 10356334, 37205121, 134887153, 493000086, 1814902409, 6724595543, 25061885217, 93899071368, 353514105817, 1336822098961, 5075833932200
OFFSET
0,4
COMMENTS
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.
FORMULA
Euler transform of b(1) = 0, b(n > 1) = A035053(n). - Gus Wiseman, May 21 2018
EXAMPLE
From Gus Wiseman, May 21 2018: (Start)
Non-isomorphic representatives of the a(5) = 11 hyperforests are the following:
{{1,2,3,4,5}}
{{1,2},{3,4,5}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,2},{3,5},{4,5}}
{{1,4},{2,5},{3,4,5}}
{{1,5},{2,5},{3,4,5}}
{{1,3},{2,4},{3,5},{4,5}}
{{1,4},{2,5},{3,5},{4,5}}
{{1,5},{2,5},{3,5},{4,5}}
(End)
MATHEMATICA
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];
b[0] = 0; b[n_] := b[n] = etr[etr[b]][n-1];
c[1] = 0; c[n_] := b[n] + etr[b][n] - Sum[b[k]*etr[b][n-k], {k, 0, n}];
a = etr[c];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jul 12 2018, after Alois P. Heinz's code for A035053 *)
PROG
(PARI) \\ here b is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
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
CROSSREFS
Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144958 (unlabeled forests without isolated vertices), A144959, A304716, A304717, A304867, A304911.
Sequence in context: A059075 A345882 A121134 * A131347 A292210 A079225
KEYWORD
nonn
AUTHOR
Washington Bomfim, Sep 27 2008
STATUS
approved