A392966 Number of ordered rooted trees with non-root node weights summing to n. Such that non-root nodes are each labeled with a subset s of [n], the weight of a node is the number of elements in s, only non-root nodes labeled with singleton subsets can have child nodes, and the labels of a given tree form a set partition of [n].

1, 1, 5, 40, 475, 7556, 151179, 3650774, 103370299, 3359291944, 123262552723, 5041145001410, 227402683183251, 11217008823919796, 600652993489210315, 34702022004968264638, 2151603180503209400395, 142507476123814838856608, 10041957995899619638273827

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f.: A(x) satisfies A(x) = 1 + A(x)*(exp(x) + x*A(x) - x - 1).

E.g.f.: 2 / (2+x-exp(x) + sqrt((2+x-exp(x))^2 - 4*x)). - Seiichi Manyama, Mar 12 2026

Example

a(2) = 5 counts:
    o        o         o       o     o
    |       / \       / \      |     |
  {1,2}   {1} {2}   {2} {1}   {1}   {2}
                               |     |
                              {2}   {1}

Prog

(PARI) D_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace((-(exp(x)-2-x) - sqrt((exp(x)-2-x)^2-4*x))/(2*x)))}

Crossrefs

Cf. A000108, A000670, A052895.

Sequence in context: A385585 A090362 A201366 * A281160 A282476 A198247

Adjacent sequences: A392963 A392964 A392965 * A392967 A392968 A392969

Keyword

nonn,easy

Author

John Tyler Rascoe, Mar 02 2026

Status

approved