A060905 - OEIS

%I #32 Oct 31 2024 22:27:38

%S 1,1,4,19,110,751,5902,52165,509588,5437729,62828306,780287839,

%T 10351912276,145944541159,2176931651546,34225419288421,

%U 565282627986368,9779830102138945,176776613812205074,3330780287838743575

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

%C 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

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

%H Alois P. Heinz, <a href="/A060905/b060905.txt">Table of n, a(n) for n = 0..200</a>

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

%F 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.

%F 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

%t 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 *)

%o (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 */

%Y Cf. A000949 - A000951, A060905 - A060913.

%Y Column k=3 of A245501.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Apr 07 2001