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%I #46 Jun 28 2021 15:59:00
%S 0,1,2,5,13,37,108,332,1042,3360,11019,36722,123875,422449,1453553,
%T 5040816,17599468,61814275,218252584,774226549,2758043727,9862357697,
%U 35387662266,127374191687,459783039109,1664042970924,6037070913558,21951214425140,79981665585029
%N Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
%H Alois P. Heinz, <a href="/A036249/b036249.txt">Table of n, a(n) for n = 0..1717</a>
%H F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%H F. Chapoton, F. Hivert, J.-C. Novelli, <a href="https://doi.org/10.1016/j.jalgebra.2016.07.001">A set-operad of formal fractions and dendriform-like sub-operads</a>, Journal of Algebra, 465 (2016), 322-355.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=768">Encyclopedia of Combinatorial Structures 768</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F G.f. satisfies: A(x) = x*exp( Sum_{n>=1} (A(x^n) + x^n)/n ). - _Paul D. Hanna_, Oct 19 2005
%F If b(n) is the Euler transform of a(n), A052855, then a(n+1) = a(n) + b(n). - _Franklin T. Adams-Watters_, Mar 09 2006
%F G.f.: (x/(1 - x)) * Product_{n>=1} 1/(1 - x^n)^a(n). - _Ilya Gutkovskiy_, Jun 28 2021
%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*
%p add(d*a(d), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n-1)) end:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jun 13 2018
%t max = 27; A[_] = 1; Do[A[x_] = x*Exp[Sum[(A[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; CoefficientList[A[x] + O[x]^max, x] (* _Jean-François Alcover_, May 25 2018 *)
%o (PARI) {a(n)=local(A=x+x*O(x^n));for(i=1,n, A=x*exp(sum(m=1,n,(subst(A,x,x^m)+x^m)/m)));polcoeff(A,n,x)} \\ _Paul D. Hanna_, Oct 19 2005
%Y Essentially the same as A029856. Cf. A048802. Row sums of A303911.
%K nonn
%O 0,3
%A _Christian G. Bower_, Nov 15 1998
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