A370949 Triangle read by rows: T(n,k) is the number of forests of labeled rooted Greg hypertrees with n white vertices and k black vertices, 0 <= k < n.

1, 3, 1, 19, 16, 3, 189, 268, 115, 15, 2576, 5221, 3655, 1050, 105, 44683, 118599, 117236, 54040, 11655, 945, 941977, 3102184, 3996384, 2581138, 883575, 152460, 10395, 23388025, 92149019, 147043422, 123318510, 58806055, 15980580, 2297295, 135135

Offset: 1

Author

Paul Laubie, Mar 06 2024

Keywords

• nonn
• tabl

Comments

A rooted Greg hypertree is a hypertree with black and white vertices such that white vertices are labeled, black vertices are unlabeled, and each black vertex has at least two children.

See A048160 for the analog sequence for Greg trees.

Examples

			Triangle T(n,k) begins:
n\k    0     1     2     3     4 ...
1      1;
2      3,    1;
3     19,   16,    3;
4    189,  268,  115,   15;
5   2576, 5221, 3655, 1050,  105;
...
		

Links

• Paul Laubie, Hypertrees and embedding of the FMan operad, arXiv:2401.17439 [math.QA], 2024.

Crossrefs

Cf. A048160, A052888 (k=0), A001147 (k=n-1).

Row sums are A364816.

Programs

PARI

T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+x) - y*exp(x) + y*x + y)*exp(-x) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 06 2024

Formula

E.g.f.: series reversion in t of (log(1+t) - u*exp(t) + u*t + u)*exp(-t), where the formal variable u encodes the number of black vertices.

T(n,0) = A052888(n).

T(n,n-1) = A001147(n).