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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.
4
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
1,2
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
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.
EXAMPLE
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;
...
PROG
(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
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.
Sequence in context: A228721 A108290 A308804 * A108291 A019615 A132744
KEYWORD
nonn,tabl,easy
AUTHOR
Paul Laubie, Oct 20 2023
EXTENSIONS
a(23) corrected by Andrew Howroyd, Jan 01 2024
STATUS
approved