A060905 Expansion of e.g.f. exp(x*exp(x) + 1/2*x^2*exp(x)^2).

1, 1, 4, 19, 110, 751, 5902, 52165, 509588, 5437729, 62828306, 780287839, 10351912276, 145944541159, 2176931651546, 34225419288421, 565282627986368, 9779830102138945, 176776613812205074, 3330780287838743575

Offset: 0

Author

Vladeta Jovovic, Apr 07 2001

Keywords

• nonn

Comments

Number of functions f from a set of size n to itself such that f(f(f(x))) = f(x). - Joel B. Lewis, Dec 12 2011

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Crossrefs

Cf. A000949 - A000951, A060905 - A060913.

Column k=3 of A245501.

Programs

Mathematica

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 18 2012 *)

Maxima

a(n):=sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Aug 20 2010 */

Formula

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 2.

a(n) = sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n), n>0. - Vladimir Kruchinin, Aug 20 2010