A318494 Number of rooted simple connected graphs on n unlabeled nodes where every block is a complete graph with nonroot nodes of two colors.

1, 2, 10, 50, 285, 1696, 10647, 68842, 456922, 3091546, 21252396, 147992264, 1041779912, 7401119718, 52996414666, 382095695324, 2771458821772, 20209364313202, 148064910503435, 1089415620952020, 8046283404651000, 59635009544475814, 443380411766040664

Offset: 1

Author

Andrew Howroyd, Aug 27 2018

Keywords

• nonn

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.

Examples

			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
   \ /     \ /     \
    *       *       *
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1116 (first 200 terms from Andrew Howroyd)

Crossrefs

Cf. A007563, A134959.

Programs

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 *)

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}