A048802 Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
1, 3, 16, 133, 1521, 22184, 393681, 8233803, 198342718, 5408091155, 164658043397, 5537255169582, 203840528337291, 8153112960102283, 352079321494938344, 16325961781591781401, 809073412162081974237, 42674870241038732398720, 2386963662244981472850709
Offset: 1
Keywords
Examples
G.f. = x + 3*x^2 + 16*x^3 + 133*x^4 + 1521*x^5 + 22184*x^6 + 393681*x^7 + ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
- F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 861
- B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
- Index entries for sequences related to rooted trees
Programs
-
Mathematica
nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ ComposeSeries[ Series[t,{x,0,nn}],Series[Exp[x]-1 ,{x,0,nn}]],x] (* Geoffrey Critzer, Sep 16 2012 *) -
PARI
{a(n) = sum( k=1, n, stirling(n, k, 2) * k^(k - 1))}; /* Michael Somos, Jun 09 2012 */ -
PARI
{a(n) = n! * polcoeff( serreverse( log(1 + x*exp(-x +x*O(x^n))) ),n)} for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Jan 24 2016
Formula
E.g.f.: B(exp(x)-1) where B is e.g.f. of A000169.
E.g.f.: Series_Reversion( log(1 + x*exp(-x)) ). - Paul D. Hanna, Jan 24 2016
a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003
Stirling transform of A000169. - Michael Somos, Jun 09 2012
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n) * (log(1+exp(-1)))^(n-1/2)). - Vaclav Kotesovec, Feb 17 2014