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

1,2

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A036249, A038052, A058863, A052807.

Sequence in context: A241464 A341852 A141628 * A213357 A119392 A307979

Adjacent sequences: A048799 A048800 A048801 * A048803 A048804 A048805

Keyword

nonn

Author

Christian G. Bower, Mar 15 1999

Status

approved