#C27.txt: May 1, 2025
Help27:=proc(): print(` NuULGe(N,r), LkT(k,N),LkUT(k,N) `):end:
read `AGT.txt`:

#NuULGe(N,r): The first N terms of the sequence enumerating UNLABELED GRAPHS with n vertices
#and n+r edges
NuULGe:=proc(N,r) local L,x,i,t:
L:=[seq(CIPknC(i,x),i=1..N)]:
L:=subs({seq(x[i]=1+t^i,i=1..binomial(N,2))},L):
[seq(coeff(L[i],t, i+r ), i=1..N)]:
end:


#LkT(k,N): the first N terms of the number of ORDERED full k-ary trees with (k-1)*n+1 leaves
LkT:=proc(k,N) local x,f,i,i1:
f := x:

for i from 1 to (k-1)*N+1 do 
f := expand(x+f^k);
 f := add(coeff(f,x,i1)*x^i1,i1=1..(k-1)*N+1) :
od:
[seq(coeff(f,x, (k-1)*i+1),i=1..N)] :
end:

#added after class

#LkUT(k,N): the first N terms of the number of UNORDERED full k-ary trees with (k-1)*n+1 leaves
LkUT:=proc(k,N) local x,f,i,F,X,i1; 
F:=CIPsn(k,X):
f:=x:
for i from 1 to (k-1)*N+1 do
f:=x+subs({seq(X[i1]=subs(x=x^i1,f),i1=1..k)},F):
f:= add(coeff(f,x,i1)*x^i1,i1=1..(k-1)*N+1):
od:
[seq(coeff(f,x,(k-1)*i+1),i=1..N)] :
end: