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

1,2

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.

Links

Table of n, a(n) for n=1..36.

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

Example

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;
...

Prog

(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

Crossrefs

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

Row sums are A364816.

Sequence in context: A372175 A027537 A192721 * A002380 A274075 A374651

Adjacent sequences: A370946 A370947 A370948 * A370950 A370951 A370952

Keyword

nonn,tabl

Author

Paul Laubie, Mar 06 2024

Status

approved