A318494 Number of rooted 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

1,2

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

Cf. A007563, A134959.

Sequence in context: A026332 A027908 A206637 * A020088 A205772 A306336

Adjacent sequences: A318491 A318492 A318493 * A318495 A318496 A318497

Keyword

nonn

Author

Andrew Howroyd, Aug 27 2018

Status

approved