A036249 Number of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
0, 1, 2, 5, 13, 37, 108, 332, 1042, 3360, 11019, 36722, 123875, 422449, 1453553, 5040816, 17599468, 61814275, 218252584, 774226549, 2758043727, 9862357697, 35387662266, 127374191687, 459783039109, 1664042970924, 6037070913558, 21951214425140, 79981665585029
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1717
- Håvard Berland, Brynjulf Owren and Bård Skaflestad, B-series and order conditions for exponential integrators, 2004. See p. 6.
- F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
- F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, Journal of Algebra, 465 (2016), 322-355.
- Timothy Y. Chow and Mark G. Tiefenbruck, The Latin Tableau Conjecture, 2024. See p. 11.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 768
- Index entries for sequences related to rooted trees
Programs
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Maple
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)* add(d*a(d), d=numtheory[divisors](j)), j=1..n)/n) end: a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n-1)) end: seq(a(n), n=0..35); # Alois P. Heinz, Jun 13 2018 -
Mathematica
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 *)
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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
Formula
G.f. satisfies: A(x) = x*exp( Sum_{n>=1} (A(x^n) + x^n)/n ). - Paul D. Hanna, Oct 19 2005
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
G.f.: (x/(1 - x)) * Product_{n>=1} 1/(1 - x^n)^a(n). - Ilya Gutkovskiy, Jun 28 2021