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: