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

%I A318494 #30 Jan 29 2025 11:24:22
%S A318494 1,2,10,50,285,1696,10647,68842,456922,3091546,21252396,147992264,
%T A318494 1041779912,7401119718,52996414666,382095695324,2771458821772,
%U A318494 20209364313202,148064910503435,1089415620952020,8046283404651000,59635009544475814,443380411766040664
%N A318494 Number of rooted simple connected graphs on n unlabeled nodes where every block is a complete graph with nonroot nodes of two colors.
%C A318494 Number of rooted spanning hypertrees on n unlabeled nodes with edges of size 1 allowed.
%C A318494 Shifts left when Euler transform is applied twice to double this sequence.
%H A318494 Alois P. Heinz, <a href="/A318494/b318494.txt">Table of n, a(n) for n = 1..1116</a> (first 200 terms from Andrew Howroyd)
%e A318494 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.
%e A318494   o---o   o   o   o---o
%e A318494    \ /     \ /     \
%e A318494     *       *       *
%p A318494 b:= ((proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
%p A318494         add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p A318494       end end)@@2)(2*a):
%p A318494 a:= n-> b(n-1):
%p A318494 seq(a(n), n=1..25);  # _Alois P. Heinz_, Aug 27 2018
%t A318494 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];
%t A318494 a[n_] := b[n-1];
%t A318494 b = etr@etr@(2a[#]&);
%t A318494 Array[a, 25] (* _Jean-François Alcover_, Nov 01 2020 *)
%o A318494 (PARI)
%o A318494 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A318494 seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
%Y A318494 Cf. A007563, A134959.
%K A318494 nonn
%O A318494 1,2
%A A318494 _Andrew Howroyd_, Aug 27 2018