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A036249 Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

%I A036249 #56 Nov 11 2024 14:38:47
%S A036249 0,1,2,5,13,37,108,332,1042,3360,11019,36722,123875,422449,1453553,
%T A036249 5040816,17599468,61814275,218252584,774226549,2758043727,9862357697,
%U A036249 35387662266,127374191687,459783039109,1664042970924,6037070913558,21951214425140,79981665585029
%N A036249 Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
%H A036249 Alois P. Heinz, <a href="/A036249/b036249.txt">Table of n, a(n) for n = 0..1717</a>
%H A036249 Håvard Berland, Brynjulf Owren and Bård Skaflestad, <a href="https://cds.cern.ch/record/848116/files/cer-002531450.pdf">B-series and order conditions for exponential integrators</a>, 2004. See p. 6.
%H A036249 F. Chapoton, F. Hivert, and 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 A036249 F. Chapoton, F. Hivert, and 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 A036249 Timothy Y. Chow and Mark G. Tiefenbruck, <a href="https://timothychow.net/latin11.pdf">The Latin Tableau Conjecture</a>, 2024. See p. 11.
%H A036249 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=768">Encyclopedia of Combinatorial Structures 768</a>
%H A036249 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A036249 G.f. satisfies: A(x) = x*exp( Sum_{n>=1} (A(x^n) + x^n)/n ). - _Paul D. Hanna_, Oct 19 2005
%F A036249 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 A036249 G.f.: (x/(1 - x)) * Product_{n>=1} 1/(1 - x^n)^a(n). - _Ilya Gutkovskiy_, Jun 28 2021
%p A036249 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*
%p A036249       add(d*a(d), d=numtheory[divisors](j)), j=1..n)/n)
%p A036249     end:
%p A036249 a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n-1)) end:
%p A036249 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jun 13 2018
%t A036249 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 A036249 (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 A036249 Essentially the same as A029856. Cf. A048802. Row sums of A303911.
%K A036249 nonn
%O A036249 0,3
%A A036249 _Christian G. Bower_, Nov 15 1998