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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
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
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
KEYWORD
nonn,tabl
AUTHOR
Paul Laubie, Mar 06 2024
STATUS
approved