A318607 Triangle read by rows: T(n,k) is the number of sets of rooted hypertrees on a total of n unlabeled nodes with a total of k edges, (0 <= k < n).

1, 1, 1, 1, 2, 2, 1, 3, 6, 4, 1, 4, 12, 16, 9, 1, 5, 20, 42, 46, 20, 1, 6, 30, 86, 145, 128, 48, 1, 7, 42, 153, 353, 483, 364, 115, 1, 8, 56, 248, 729, 1369, 1592, 1029, 286, 1, 9, 72, 376, 1345, 3236, 5150, 5151, 2930, 719, 1, 10, 90, 541, 2287, 6728, 13708, 18792, 16513, 8344, 1842

Offset: 1

Author

Andrew Howroyd, Aug 30 2018

Keywords

• nonn
• tabl

Comments

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

Bivariate Euler transform of triangle A318602.

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 6, 4;
  1, 4, 12, 16, 9;
  1, 5, 20, 42, 46, 20;
  1, 6, 30, 86, 145, 128, 48;
  1, 7, 42, 153, 353, 483, 364, 115;
  1, 8, 56, 248, 729, 1369, 1592, 1029, 286;
  ...
Case n=3: There are 5 sets of rooted graph which are illustrated below (an x marks a root node). These have 0, 1, 1, 2, 2 blocks so row 3 is 1, 2, 2.
      x        o        o        o        o
              /        / \        \      /
    x   x    x   x    x---o    x---o    x---o
		

Links

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

Crossrefs

Rightmost diagonal is A000081 (rooted trees).

Row sums are A035052.

Cf. A318601, A318602.

Programs

PARI

\\ here EulerMT is Euler transform (bivariate version).
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)}
A(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerMT(y*EulerMT(v)))); [Vecrev(p) | p <- EulerMT(v)]}
{ my(T=A(10)); for(n=1, #T, print(T[n])) }