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29 December

Counting Binary Trees (The Hard Way)

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

9 November
Posted in Article, Math Problems

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?

6 November

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 .

14 August
Posted in Article, Math Problems

Even Pascals

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

14 August
Posted in Article, Math Problems

Sequences in Sequences

Define the sequence recursively by and Find an explicit formula for in terms of n.

14 August
Posted in Article, Math Problems

More Linear Recurrences

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

13 August
Posted in Article, Math Problems

Almost Linear

Find the general term of the sequence given by , and 13 August
Posted in Article, Math Problems

Polynomial Divisors

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

13 August
Posted in Article, Math Problems

Analytical Fibonacci

We derive Binet’s equation for the nth Fibonacci number as 13 August
Posted in Article, Math Problems

Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a kth linear recursive sequence is defined as 