A134956 Number of hyperforests with n labeled vertices: analog of A134954 when edges of size 1 are allowed (with no two equal edges).

1, 2, 8, 64, 880, 17984, 495296, 17255424, 728771584, 36208782336, 2069977144320, 133869415030784, 9664049202221056, 770400218809384960, 67219977066339008512, 6372035504466437079040, 652103070162164448952320, 71656927837957783339925504

Offset

0,2

References

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Links

Alois P. Heinz, Table of n, a(n) for n = 0..335

Formula

Equals 2^n*A134954(n).

a(n) = Sum of n!prod_{k=1}^n\{ frac{ A134958(k)^{c_k} }{ k!^{c_k} c_k! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008

Example

From Gus Wiseman, May 21 2018: (Start)
The a(2) = 8 hyperforests are the following:
  {{1},{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1,2}}
  {{1},{2}}
  {{1}}
  {{2}}
  {}
(End)

Maple

with(combinat): p:= proc(n) option remember; add(stirling2(n-1, i) *n^(i-1), i=0..n-1) end: g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: a:= n-> `if`(n=0, 1, 2^n * g(n)): seq(a(n), n=0..30); # Alois P. Heinz, Oct 07 2008

Mathematica

p[n_] := p[n] = Sum[ StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; g[n_] := g[n] = p[n] + Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; a[n_] := If[n == 0, 1, 2^n* g[n]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Crossrefs

Cf. A134958. - Washington Bomfim, Sep 25 2008

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A304716, A304717, A304867, A304911.

Sequence in context: A153560 A321059 A331957 * A011803 A007625 A085658

Adjacent sequences: A134953 A134954 A134955 * A134957 A134958 A134959

Keyword

nonn

Author

Don Knuth, Jan 26 2008

Status

approved