A392966 - OEIS

%I #33 Mar 12 2026 11:36:42

%S 1,1,5,40,475,7556,151179,3650774,103370299,3359291944,123262552723,

%T 5041145001410,227402683183251,11217008823919796,600652993489210315,

%U 34702022004968264638,2151603180503209400395,142507476123814838856608,10041957995899619638273827

%N 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].

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

%F E.g.f.: 2 / (2+x-exp(x) + sqrt((2+x-exp(x))^2 - 4*x)). - _Seiichi Manyama_, Mar 12 2026

%e a(2) = 5 counts:

%e     o        o         o       o     o

%e     |       / \       / \      |     |

%e   {1,2}   {1} {2}   {2} {1}   {1}   {2}

%e                                |     |

%e                               {2}   {1}

%o (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)))}

%Y Cf. A000108, A000670, A052895.

%K nonn,easy

%O 0,3

%A _John Tyler Rascoe_, Mar 02 2026