Help:=proc(): print(`LIF(P,u,N) , URT(N) `): end:
#LIF: Inputs an expression P of the variable
#u that possesses
#a Maclaurin expansion in u, and outputs the
#first N terms of the series expansion of the
#unique f.p.s u(t) satisfying the functional eq.
#u(t)=t*P(u(t)). For example,
#LIF(1+u^2,u,10); should return something with
#Catalan numbers
LIF:=proc(P,u,N) local n:
[seq(coeff(taylor(P^n, u=0,n),u,n-1)/n,n=1..N)]:
end:
#Counting unlabelled rooted trees
#A[n]=number of rooted unalabelled trees with n vertices
#A[1]=1, A[2]=1, A[3]=2,
#Removing the root, leads to a MULTISET of smaller rooted trees
#HOW MANY with one vertex? 1/(1-x)^A[1]
#HOW MANY with two vertices 1/(1-x^2)^A[2]
#...
#How MANY with i vertices? 1/(1-x^i)^A[i]
#A[1]*x+A[2]*x^2+A[3]*x^3+A[4]*x^4+...=
#x/(1-x)^A[1]/(1-x^2)^A[2]/(1-x^3)^A[3]/.....
#URT(N): the first N terms of the enumerating
#sequence for the numnber of UNLABELLED rooted trees
#with n vertices
URT:=proc(N) local A,f,n,x,i:
A:=[1]:
for n from 2 to N do
f:=x*mul(1/(1-x^i)^A[i],i=1..nops(A)):
f:=taylor(f,x=0,n+1):
A:=[op(A),coeff(f,x,n)]:
od:
A:
end: