A007563 -id:A007563 - cp's OEIS frontend

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

1, 1, 2, 4, 8, 17, 39, 98, 263, 759, 2299, 7259, 23649, 79057, 269629, 935328, 3290260, 11714285, 42139053, 152963037, 559697097, 2062574000, 7649550572, 28534096988, 106994891146, 403119433266, 1525466082179, 5795853930652, 22102635416716, 84579153865570

Offset: 0

Gus Wiseman, May 22 2018

nonn

			Non-isomorphic representatives of the a(4) = 8 hypertrees are the following:
{}
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ 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)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Partial sums of A035053 if we assume A035053(1) = 0.

a(n) = A304937(n) + 1 for n > 0.

A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

Original entry on oeis.org

1, 0, 1, 3, 7, 16, 38, 97, 262, 758, 2298, 7258, 23648, 79056, 269628, 935327, 3290259, 11714284, 42139052, 152963036, 559697096, 2062573999, 7649550571, 28534096987, 106994891145, 403119433265, 1525466082178, 5795853930651, 22102635416715, 84579153865569

Offset: 0

Gus Wiseman, May 21 2018

nonn

			Non-isomorphic representatives of the a(5) = 16 hypertrees are the following:
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,2,3,4,5}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,5},{2,3,4,5}}
{{1,2,5},{3,4,5}}
{{1,2},{2,5},{3,4,5}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{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}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304867, A304911, A304912, A304918.

\\ 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)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)) - x)/(1-x))} \\ Andrew Howroyd, Aug 27 2018

a(n) = a(n-1) + A035053(n) for n > 1, a(n) = 1 - n for n < 2.

A318602 Triangle read by rows: T(n,k) is the number of rooted hypertrees on n unlabeled nodes with k edges, (0 <= k < n).

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 4, 0, 1, 5, 10, 9, 0, 1, 6, 20, 30, 20, 0, 1, 8, 33, 77, 91, 48, 0, 1, 9, 49, 152, 277, 268, 115, 0, 1, 11, 68, 269, 655, 969, 790, 286, 0, 1, 12, 91, 428, 1330, 2651, 3294, 2308, 719, 0, 1, 14, 116, 647, 2420, 6137, 10300, 10993, 6737, 1842

Offset: 1

Andrew Howroyd, Aug 29 2018

Equivalently, the number of rooted connected graphs on n unlabeled nodes with k blocks where every block is a complete graph.

			Triangle begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  3,  4;
  0, 1,  5, 10,   9;
  0, 1,  6, 20,  30,  20;
  0, 1,  8, 33,  77,  91,  48;
  0, 1,  9, 49, 152, 277, 268, 115;
  0, 1, 11, 68, 269, 655, 969, 790, 286;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Rightmost diagonal is A000081 (rooted trees).
Row sums are A007563.
Cf. A318601.

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
R(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- v]}
{ my(T=R(10));for(n=1, #T, print(T[n])) }

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

Original entry on oeis.org

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

Washington Bomfim, Sep 28 2008

nonn

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.

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

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

Cf. A134955, A035053 (hypertrees).

\\ 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

cp's OEIS Frontend

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

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A304937 Number of unlabeled nonempty hypertrees with up to n vertices and no singleton edges.

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A318602 Triangle read by rows: T(n,k) is the number of rooted hypertrees on n unlabeled nodes with k edges, (0 <= k < n).

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

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