OFFSET
1,2
COMMENTS
Number of rooted spanning hypertrees on n unlabeled nodes with edges of size 1 allowed.
Shifts left when Euler transform is applied twice to double this sequence.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1116 (first 200 terms from Andrew Howroyd)
EXAMPLE
a(3) = 10 because there are three possible rooted graphs which are illustrated below and these can be colored up to isomorphism in 3, 3 and 4 ways respectively.
o---o o o o---o
\ / \ / \
* * *
MAPLE
b:= ((proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
end end)@@2)(2*a):
a:= n-> b(n-1):
seq(a(n), n=1..25); # Alois P. Heinz, Aug 27 2018
MATHEMATICA
etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
a[n_] := b[n-1];
b = etr@etr@(2a[#]&);
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Aug 27 2018
STATUS
approved