Help:=proc(): print(` GuessAlg(L,P,x,MaxDegP) `): 
print( ` GuessAlg1(L,P,x,DegP) , GuessAlg11(L,P,x,DegP,Degx) `):

end:
#C10.txt: starting the Alg. Formal Power Series
#1+1!*x+2!*x^2+3!*x^3+ ......

#GuessAlg(L,P,x,MaxDegP): Inputs a sequence of numbers L
#and tries to guess  an alg. eq. of the from
#F(P,x)=0. where P is the symbol for the generating
#function of L, P(x)=add(L[i+1]*x^i,i=0..infinity)
#of max. degree MaxDegP
GuessAlg:=proc(L,P,x, MaxDegP) local g,DegP:

for DegP from 1 to MaxDegP do
 g:=GuessAlg1(L,P,x,DegP):
 if g<>FAIL then
     RETURN(g):
 fi:
od:

FAIL:

end:


#GuessAlg1(L,P,x,DegP): Inputs a sequence of numbers L
#and tries to guess  an alg. eq. of the from
#F(P,x)=0. where P is the symbol for the generating
#function of L, P(x)=add(L[i+1]*x^i,i=0..infinity)
#of degree DegP in P
GuessAlg1:=proc(L,P,x, DegP) local g,Degx:



for Degx from 0 while (1+Degx)*(1+DegP)<nops(L)-6 do

g:=GuessAlg11(L,P,x, DegP,Degx):

if g<>FAIL then
  RETURN(g):
fi:

od:

FAIL:

end:



#GuessAlg11(L,P,x,DegP, Degx): Inputs a sequence of numbers L
#and tries to guess  an alg. eq. of the from
#F(P,x)=0. where P is the symbol for the generating
#function of L, P(x)=add(L[i+1]*x^i,i=0..infinity)
#of degree DegP in P and Degx in x
GuessAlg11:=proc(L,P,x, DegP, Degx) local g, F,a, i,j,
var,P1,F1,eq, var1,v:

F:=add(add(a[i,j]*x^i*P^j,i=0..Degx),j=0..DegP):

var:={seq(seq(a[i,j],i=0..Degx),j=0..DegP)}:

P1:=add(L[i]*x^(i-1),i=1..nops(L)):

F1:=expand(subs(P=P1,F)):

eq:={seq( coeff(F1,x,i)=0, i=0..nops(L)-1)}:

var1:=solve(eq,var):

F:=subs(var1,F):

if F=0 then
 RETURN(FAIL):
else
F:=subs({seq(v=1, v  in var)},F):
 
F:=add(factor(coeff(F,P,i))*P^i,i=0..degree(F,P)): 
RETURN(F):
fi:



end: