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Patrick B.

• 16

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Give a closed formula for C(n).

C(n) = 1/(n+1) * nCr(2n, n)

Give a Recurrence Relation for the Catalan Numbers involving a Summation.

C(0) = 1,

C(n+1) = Sum(i=0, n, C(i)*C(n-i))

for n >= 0.

C(n+1) = Sum(i=0, n, C(i)*C(n-i))

for n >= 0.

Give a Recurrence Relation for the Catalan Numbers NOT involving a Summation.

C(0) = 1,

C(n+1) = 2*(2n+1) / (n+2) * C(n)

C(n+1) = 2*(2n+1) / (n+2) * C(n)

In terms of the Fibonacci Numbers, What is Pascal's Relation?

Prove by showing h(n) = h(n-1) + h(n-2).

Prove by showing h(n) = h(n-1) + h(n-2).

for N>=1,

F(n) = Sum(k=0, n-1,

nCr(n-k-1, k))

F(n) = Sum(k=0, n-1,

nCr(n-k-1, k))

Give the Recurrence Relation for the Fibonacci Numbers.

What type of linear recurrence relation?

What type of linear recurrence relation?

F(0) = 0, F(1) = 1,

F(n) = F(n-1) + F(n-2)

This is a "Second Degree" Linear Recurrence Relation.

F(n) = F(n-1) + F(n-2)

This is a "Second Degree" Linear Recurrence Relation.

Give an exact formula for the Fibonacci Numbers.

F(n) = 1/sqrt(5) * ( ( (1+sqrt(5))/2 )^n - ( (1-sqrt(5))/2 )^n )

Give the Generating Function for the Fibonacci Numbers.

Define a K-term Linear Recurrence Relation.

Give the general generating function for a K-term Linear Recurrence Relation.

Explain how to solve a homogeneous linear recurrence relation.

Explain how to solve a non-homogeneous, non-linear recurrence relation.

Give the recurrence relation for the Derangement Sequence.

What Proportion of Permutations are Derangements?

Give the Generating Function for the Derangement Sequence.

Give the General Form of a generating function.

The End

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