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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.

%I A364709 #27 Jan 01 2024 19:44:23
%S A364709 1,2,1,9,9,1,64,96,28,1,625,1250,625,75,1,7776,19440,14040,3240,186,1,
%T A364709 117649,352947,336140,120050,14749,441,1,2097152,7340032,8716288,
%U A364709 4300800,870912,61824,1016,1,43046721,172186884,245525742,156243654,45605511,5664330,245025,2295,1
%N 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.
%C A364709 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 A364709 T(n,k) is also the dimension of the operad ComPreLie in arity n with k commutative products.
%H A364709 Andrew Howroyd, <a href="/A364709/b364709.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A364709 E.g.f: series reversion in t of (log(1+x*t)/x)*exp(-t).
%F A364709 T(n,0) = n^(n-1).
%F A364709 T(n,n-1) = 1.
%e A364709 Triangle T(n,k) begins:
%e A364709 n\k   0     1    2    3    4 ...
%e A364709 1     1;
%e A364709 2     2,    1;
%e A364709 3     9,    9,   1;
%e A364709 4    64,   96,  28,   1;
%e A364709 5   625, 1250, 625,  75,   1;
%e A364709 ...
%o A364709 (PARI) T(n) = my(x='x+O('x^(n+1))); [Vecrev(p) | p<-Vec(serlaplace( serreverse(log(1+x*y)*exp(-x)/y )))]
%o A364709 {my(A=T(10)); for(n=1, #A, print(A[n]))} \\ _Andrew Howroyd_, Oct 20 2023
%Y A364709 Cf. A000169 (k=0), A081131 (k=1).
%Y A364709 Row sums are A052888.
%Y A364709 Series reversion as e.g.f of A111492 with an offset of 1.
%K A364709 nonn,tabl,easy
%O A364709 1,2
%A A364709 _Paul Laubie_, Oct 20 2023
%E A364709 a(23) corrected by _Andrew Howroyd_, Jan 01 2024