#its ok to post
#TaerimKim,11/15/2020,Assignment 19

#1.
#The algebraic method of proving the egf(C) = egf(A)egf(B) when C = A x B seems very logical
#and mathematically simplified to make easier sense. 

#The way I understood the concept was that the condition A xB = C where every labels are disjoint and distinct
#played a important role because that would mean that C is a integrated set of A and B without repetition.
#So technically efg(C) is that you are picking n labels from the pool that combined A and B and order them in permutation
#In other words, i can pick k labels from elements of A pool and hence n-k label from that of B pool 
#= picking k element from A pool x picking n-k element from B pool = egf(A) x egf(B)
#
#The definition of c(n) helps this understanding as it mathematically designs all the possibilities from picking none of A components to all A components (k=0..n)
#hence, from n we pick k element from A and there are binomial(n,k) ways of placing a k label among n then all the possibility of choosing the other n-k component from b 
#So then, from the mathematical proof we can see that the idea of egf convolution of two series yielding efg of convoluted sum of the original sequences.


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#2. 

#a(n)=n*(n-1)*(n-2) (n กร 0)

#i)OGF
#f(x) = Sum(a(n)*x^n,n=0..inf)
#a(0)=0; a(1)=0; a(2)=0; a(3) = 3*2*1; a(4)=4*3*2; a(5)=5*4*3; 
# a(n) = n!/(n-3)!

#so the ogf is f(x) = 0*x^0 + 0*x^1 + 0*x^2 + 3!/0!*x^3 + 4!/1!x^4 + 5!/2!x^5 ... + n!/(n-3)!x^n

#ii)egf
#this is basically a(n)/n! = n!/(n-3)!n! = 1/(n-3)!
# So egf would be f(x) = (0/0!)*x^0 + (0/1!)*x^1 + (0/2!)*x^2 + (3!/1!3!)*x^3 + (4!/1!4!)x^4 + (5!/2!*5!)x^5 ... + n!/(n-3)!n!x^n 

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#3. By the fundamental theorem 
#egf(SP1) = exp(exp(x)-1)
#egf(SP2) = exp(exp(x)-1)

#egf(k) = egf(SP1)*egf(SP2) = exp^2(exp(x)-1)
#(i) coeff(taylor(exp^2(exp(x)-1),x=0,101)
exp^2/93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
#(i)coeff(taylor(exp^2(exp(x)-1),x=0,101) * 100!
exp^2