A035052 - OEIS

%I #35 May 21 2018 02:58:09

%S 1,1,2,5,14,42,134,444,1518,5318,18989,68856,252901,938847,3517082,

%T 13278844,50475876,193014868,741963015,2865552848,11113696421,

%U 43266626430,169019868095,662337418989,2602923589451,10256100717875

%N Number of sets of rooted connected graphs where every block is a complete graph.

%H T. D. Noe, <a href="/A035052/b035052.txt">Table of n, a(n) for n=0..200</a>

%H Loic Foissy, <a href="https://hal.archives-ouvertes.fr/hal-00808513">The Hopf algebra of Fliess operators and its dual pre-Lie algebra</a>, 2013.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=862">Encyclopedia of Combinatorial Structures 862</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Euler transform of A007563.

%F a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.35683683547585... . - _Vaclav Kotesovec_, Jul 26 2014

%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(aa): c:= etr(b): aa:= n-> if n=0 then 0 else c(n-1) fi: a:= etr(aa): seq(a(n), n=0..25); # _Alois P. Heinz_, Sep 09 2008

%t 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]; b = etr[aa]; c = etr[b]; aa = Function[{n}, If[n == 0, 0, c[n-1]]]; a = etr[aa]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 05 2015, after _Alois P. Heinz_ *)

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([1], EulerT(v))} \\ _Andrew Howroyd_, May 20 2018

%Y Cf. A007549, A007563, A030019, A035051, A035053.

%Y Cf. A245566.

%K nonn

%O 0,3

%A _Christian G. Bower_, Oct 15 1998