#OK to post homework
#Ariana Yousafzai, 10/18/2020, Assignment 11


1. 11 dolls in 5 towers → 246730
11 dolls in any number of towers → SP(11) in Maple → must be a very, very large number since this command caused my computer to crash 


2. xn := proc(x, n) sum(x(n - 1), i = 0 .. n); end proc


Axn := proc(x, n) sum(Snk(n, k)*xn(x, k), i = 0 .. n); end proc
3.  
c(n-1, k-1) + c(n-1, k) = (n-1)! / (n-k)!(k-1)! + (n-1)! / (n-k-1)!(k)!
= (n-1)! / (n-k-1)!(k-1)! * (1/n-k + 1/k) 
= n! / (n-k)!(k)!
=c(n,k) 
4. cnk := proc(n, k) option remember; 
if n <= 1 then 
if 1 <= k and k <= n then 
RETURN(1); 
else 
RETURN(0); 
end if; 
end if; 
cnk(n - 1, k - 1) + k*cnk(n - 1, k); 
end proc


5. sum(c(n, k)*x^k, k = 1 .. n)