In this post, we prove the closed form of a nonlinear recurrence corresponding to the count of binary trees with nodes.

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## Posts Tagged ‘ algebra ’

# Counting Words

Suppose that we are given the alphabet ; a word of length : , is an ordered -tuple whose elements all came from . For example, a word of length in might be the tuple , which we will hereon denote as . For some natural number , how many words of length are there that contains exactly zeros?

# Annoying Mclaurin Series

Suppose that we’re given the function , find the ordinary generating function associated with it in the form of . Furthermore, find/compute .

# Even Pascals

Let , where k is an integer. Prove that for any positive integer n the number

is divisible by .

# Sequences in Sequences

Define the sequence recursively by and

Find an explicit formula for in terms of n.

# More Linear Recurrences

Let be defined by the recurrence relation , with . Show that the expression depends only on b and , but not on a.

# Almost Linear

Find the general term of the sequence given by , and

# Polynomial Divisors

Let . Show that for any positive integers there exists unique numbers such that the polynomial is divisible by p(x)

# Analytical Fibonacci

We derive Binet’s equation for the n^{th} Fibonacci number as

# Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a k^{th} linear recursive sequence is defined as