A triangle of numbers where each number is the sum of the two numbers above it.
Factorials: The product of consecutive positive integers up to a given number.
Binomial Coefficients: The coefficient of each term in an expansion of (a+b)^n.
Combinatorics: The study of counting and arranging objects in different ways.
Triangular Numbers: The sequence of numbers formed by adding consecutive positive integers.
Fibonacci Sequence: The sequence of numbers where each number is the sum of the two preceding ones.
Summation Notation: A notation used to denote sums of finite or infinite sequences.
Pascal's Identity: An identity that relates the binomial coefficient of two adjacent rows in Pascal's Triangle.
Lucas Numbers: A sequence of numbers related to the Fibonacci sequence.
Catalan Numbers: A sequence of numbers that appear in many different counting problems.
Eulerian Numbers: A sequence of numbers that count permutations of a given type of pattern.
Stirling Numbers: A sequence of numbers that count partitions of a set into a given number of parts.
Generating Functions: A mathematical tool used to represent sequences as power series.
Wallis Product: A formula for calculating pi using products of infinite sequences.
q-binomial Coefficients: A generalization of the binomial coefficient using q-series.
Exponential Generating Functions: A type of generating function used to represent combinatorial sequences that involve exponential growth.
Pascal's Triangle itself: This is the most well-known form of Pascal's Triangle. It is constructed by starting with a 1 on the top row, and then each subsequent row is constructed by adding together the two numbers above it.
Fibonacci Triangle: The Fibonacci Triangle is derived from Pascal's Triangle by replacing each number with the sum of the two numbers above it, just as in the Fibonacci sequence.
Sierpinski Pascal's Triangle: In this pattern, the odd numbers are replaced with 0's and the even numbers are replaced with 1's except for the ones in the first two rows. This forms the basis for a fractal called the Sierpinski Gasket.
Lucas Triangle: The Lucas Triangle is derived from Pascal's Triangle, but instead of using the values of the Fibonacci sequence to generate the rows, we use the Lucas sequence.
Stirling Triangle: Also known as the Stirling Numbers of the Second Kind, this triangle gives the number of ways to partition a set of n elements into k non-empty subsets.
Lah Triangle: The Lah Triangle is used to calculate the number of ways to divide a set of n labeled objects into k classes.
Narayana's cows sequence: The Narayana's cows sequence is obtained from a diagonal of Pascal's Triangle by dividing each term by the previous term, beginning with the second term.