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

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

Author

Andrew Howroyd, Aug 29 2018

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Rightmost diagonal is A000081 (rooted trees).

Row sums are A007563.

Cf. A318601.

Programs

PARI

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])) }