A364709 - OEIS

%I #27 Jan 01 2024 19:44:23

%S 1,2,1,9,9,1,64,96,28,1,625,1250,625,75,1,7776,19440,14040,3240,186,1,

%T 117649,352947,336140,120050,14749,441,1,2097152,7340032,8716288,

%U 4300800,870912,61824,1016,1,43046721,172186884,245525742,156243654,45605511,5664330,245025,2295,1

%N 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.

%C 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.

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

%H Andrew Howroyd, <a href="/A364709/b364709.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

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

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

%F T(n,n-1) = 1.

%e Triangle T(n,k) begins:

%e n\k   0     1    2    3    4 ...

%e 1     1;

%e 2     2,    1;

%e 3     9,    9,   1;

%e 4    64,   96,  28,   1;

%e 5   625, 1250, 625,  75,   1;

%e ...

%o (PARI) T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]

%o {my(A=T(10)); for(n=1, #A, print(A[n]))} \\ _Andrew Howroyd_, Oct 20 2023

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

%Y Row sums are A052888.

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

%K nonn,tabl,easy

%O 1,2

%A _Paul Laubie_, Oct 20 2023

%E a(23) corrected by _Andrew Howroyd_, Jan 01 2024