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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..27.

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-Fran├žois 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: A272675 A059075 A121134 * A131347 A292210 A079225

Adjacent sequences:  A144956 A144957 A144958 * A144960 A144961 A144962

KEYWORD

nonn

AUTHOR

Washington Bomfim, Sep 27 2008

STATUS

approved

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Last modified April 16 16:18 EDT 2021. Contains 343050 sequences. (Running on oeis4.)