A048802 - OEIS

%I #47 Apr 18 2017 07:03:22

%S 1,3,16,133,1521,22184,393681,8233803,198342718,5408091155,

%T 164658043397,5537255169582,203840528337291,8153112960102283,

%U 352079321494938344,16325961781591781401,809073412162081974237,42674870241038732398720,2386963662244981472850709

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

%H Vincenzo Librandi, <a href="/A048802/b048802.txt">Table of n, a(n) for n = 1..200</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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=861">Encyclopedia of Combinatorial Structures 861</a>

%H B. R. Jones, <a href="http://summit.sfu.ca/item/14554">On tree hook length formulas, Feynman rules and B-series</a>, Master's thesis, Simon Fraser University, 2014.

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f.: B(exp(x)-1) where B is e.g.f. of A000169.

%F E.g.f.: Series_Reversion( log(1 + x*exp(-x)) ). - _Paul D. Hanna_, Jan 24 2016

%F a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-1). - _Vladeta Jovovic_, Sep 17 2003

%F Stirling transform of A000169. - _Michael Somos_, Jun 09 2012

%F a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n) * (log(1+exp(-1)))^(n-1/2)). - _Vaclav Kotesovec_, Feb 17 2014

%e G.f. = x + 3*x^2 + 16*x^3 + 133*x^4 + 1521*x^5 + 22184*x^6 + 393681*x^7 + ...

%t 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 *)

%o (PARI) {a(n) = sum( k=1, n, stirling(n, k, 2) * k^(k - 1))}; /* _Michael Somos_, Jun 09 2012 */

%o (PARI) {a(n) = n! * polcoeff( serreverse( log(1 + x*exp(-x +x*O(x^n))) ),n)}

%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 24 2016

%Y Cf. A036249, A038052, A058863, A052807.

%K nonn

%O 1,2

%A _Christian G. Bower_, Mar 15 1999