There are four types of common coins in   US   currency:

   quarters   (25 cents)
   dimes   (10 cents)
   nickels   (5 cents),   and 
   pennies   (1 cent) 


There are six ways to make change for 15 cents:

   A dime and a nickel 
   A dime and 5 pennies
   3 nickels
   2 nickels and 5 pennies
   A nickel and 10 pennies
   15 pennies


Task

How many ways are there to make change for a dollar using these common coins?     (1 dollar = 100 cents).



Optional

Less common are dollar coins (100 cents);   and very rare are half dollars (50 cents).   With the addition of these two coins, how many ways are there to make change for $1000?

(Note:   the answer is larger than   232).



References

 an algorithm from the book Structure and Interpretation of Computer Programs.
 an article in the algorithmist.
 Change-making problem on Wikipedia.


assume(p::posint,abs(x)<1):
coin:=unapply(sum(x^(p*n),n=0..infinity),p):
ways:=(amount,purse)->coeff(series(mul(coin(k),k in purse),x,amount+1),x,amount):
 
ways(100,[1,5,10,25]);
# 242
 
ways(1000,[1,5,10,25,50,100]);
# 2103596
 
ways(10000,[1,5,10,25,50,100]);
# 139946140451
 
ways(100000,[1,5,10,25,50,100]);
# 13398445413854501