A144977 a(n) = A134955(n) - A134955(n-2).

1, 1, 3, 7, 16, 41, 108, 301, 881, 2684, 8455, 27444, 91248, 309593, 1068584, 3742171, 13269281, 47561455, 172092274, 627887239, 2307902495, 8539497952, 31786480760, 118960956585, 447413177185, 1690336204778, 6412656031161

Offset

1,3

Comments

a(n) is the number of hyperforests with n unlabeled nodes without trees of order 2. This follows from the fact that for n >= 2 A134955(n-2) counts the hyperforests of order n with one or more trees of order 2.

The unique hyperforest (without loops) of order 1 is an isolated vertex, so a(1) = 1.

For n >= 2, a(n) - a(n-1) counts hyperforests of order n with components of order >= 3.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Example

a(3) = 3 since the only options are 2 hypertrees of order 3, or the forest composed by 3 isolated nodes.

Prog

(PARI) \\ here b(n) 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)); my(v=Vec(Ser(EulerT(u))*(1-x*Ser(u)))); EulerT(vector(#v, n, if(n<>2, v[n], 0)))} \\ Andrew Howroyd, Aug 27 2018

Crossrefs

Cf. A134955, A035053 (hypertrees).

Sequence in context: A323776 A036567 A018023 * A058300 A000674 A129045

Adjacent sequences: A144974 A144975 A144976 * A144978 A144979 A144980

Keyword

nonn

Author

Washington Bomfim, Sep 28 2008

Status

approved