A364709 Triangle read by rows: T(n,k) is the number of forests of labeled rooted hypertrees with n vertices and weight k, 0 <= k < n.

1, 2, 1, 9, 9, 1, 64, 96, 28, 1, 625, 1250, 625, 75, 1, 7776, 19440, 14040, 3240, 186, 1, 117649, 352947, 336140, 120050, 14749, 441, 1, 2097152, 7340032, 8716288, 4300800, 870912, 61824, 1016, 1, 43046721, 172186884, 245525742, 156243654, 45605511, 5664330, 245025, 2295, 1

Offset: 1

Author

Paul Laubie, Oct 20 2023

Keywords

• nonn
• tabl
• easy

Comments

The weight is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2.

T(n,k) is also the dimension of the operad ComPreLie in arity n with k commutative products.

Examples

			Triangle T(n,k) begins:
n\k   0     1    2    3    4 ...
1     1;
2     2,    1;
3     9,    9,   1;
4    64,   96,  28,   1;
5   625, 1250, 625,  75,   1;
...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Cf. A000169 (k=0), A081131 (k=1).

Row sums are A052888.

Series reversion as e.g.f of A111492 with an offset of 1.

Programs

PARI

T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]
{my(A=T(10)); for(n=1, #A, print(A[n]))} \\ Andrew Howroyd, Oct 20 2023

Formula

E.g.f: series reversion in t of (log(1+x*t)/x)*exp(-t).

T(n,0) = n^(n-1).

T(n,n-1) = 1.

Extensions

a(23) corrected by Andrew Howroyd, Jan 01 2024