#hw9.txt.  February 23, 2014.  Katie McKeon

read(`C9.txt`):

Help9:=proc()
print(`ApplyUmbra(P,x,U,m) PnG(n,x,U,m) GuessCoeff(n,i,U,m)`):
end:



#Inputs a polynomial P(x) and a discrete expression U(m) and
#replaces x^m by U(m) in the polynomial
ApplyUmbra:=proc(P,x,U,m) local deg, i:

add(coeff(P, x, i)*subs(m=i,U), i=0..degree(P,x)):

end:




#PnG(n,x,U,m) inputs a degree n, variable x, and discrete umbra
#expression U(m) and returns the unique monic polynomial P(x) of 
#degree n such that the umbra expression applied to P*x^i gives
#0 for all i=0,...,n-1
PnG:=proc(n,x,U,m) local P, var, eq:

P:=x^n+add(a[i]*x^i,i=0..n-1):

var:={seq(a[i],i=0..n-1)}:

eq:={ seq( ApplyUmbra(P*x^i,x,U,m), i=0..n-1)}:

sort(subs(solve(eq,var),P)):

end:


#GuessCoeff(n,i,U,m) inputs a discrete variable n, non-negative integer i, and an 
#umbra expression U(m) and tries to find a rational function R(n) so that
#R(n) gives the coefficient of x^(n-i) in PnG(n,x,U,m)
GuessCoeff:=proc(n,i,U,m) local L, j:

L:=[seq(coeff(PnG(j,x,U,m),x^(j-i)), j=i+1..10)]:
GuessRF(L,n):

end:



#Rational functions from GuessCeoff for the coefficient of x^(n-i) in PnG(n,x,U,m)
#U(m)=1/(m+1)
#i=0  ->   R(n)=1        
#i=1  ->   R(n)=-n/2-1/2
#i=2  ->   R(n)=(n^3+2+5n+4n^2)/(12+8n)
#i=3  ->   R(n)=(n^4+6+17n+17n^2+7n^3)/(-120-48n)
#i=4  ->   R(n)=(n^6+48+172n+244n^2+175n^3+67n^4+13n^5)/(3360+2304n+384n^2)
#i=5  ->   R(n)=(n^7+240+908*n+1392*n^2+1119*n^3+510*n^4+132*n^5+18*n^6)/
#		(-60480-30720*n-3840*n^2)
#i=6  ->   R(n)=(n^9+4320+18504*n+33468*n^2+33578*n^3+20643*n^4+8085*n^5+2022*n^6+312*n^7+27*n^8)/
#		(3991680+2753280*n+622080*n^2+46080*n^3)
#i=7  ->   R(n)=(n^10+30240+133848*n+252780*n^2+268514*n^3+178079*n^4+77238*n^5+22239*n^6+4206*n^7+501*n^8+34*n^9)/
#		(-103783680-57899520*n-10644480*n^2-645120*n^3)

#U(m)=1/(m+3)
#i=0  ->   R(n)=1
#i=1  ->   R(n)=(n^2+3+4*n)/(-4-2*n)
#i=2  ->   R(n)=(n^3+8+14*n+7*n^2)/(20+8*n)
#i=3  ->   R(n)=(n^4+30+61*n+41*n^2+11*n^3)/(-168-48*n)
#i=4  ->   R(n)=(n^6+432+1116*n+1104*n^2+545*n^3+143*n^4+19*n^5)/(6048+3072*n+384*n^2)
#i=5  ->   R(n)=(n^7+2520+6954*n+7585*n^2+4309*n^3+1390*n^4+256*n^5+25*n^6)/(-95040-38400*n-3840*n^2)
#i=6  ->   R(n)=(n^9+69120+218304*n+286656*n^2+208172*n^3+92904*n^4+26565*n^5+4884*n^6+558*n^7+36*n^8)/
#		(7413120+4135680*n+760320*n^2+46080*n^3)
#i=7  ->   R(n)=(n^10+544320+1789344*n+2478996*n^2+1930080*n^3+942535*n^4+303212*n^5+65303*n^6+9320*n^7+845*n^8+44*n^9)/	
#		(-172972800-81123840*n-12579840*n^2-645120*n^3)

#U(m)=eval(m!)
#i=0  ->   R(n)=1
#i=1  ->   R(n)= - computation had to be interrupted for i>0
#i=2  ->   R(n)= 
#i=3  ->   R(n)=
#i=4  ->   R(n)=
#i=5  ->   R(n)=
#i=6  ->   R(n)=
#i=7  ->   R(n)=