Triangles of numbers | Factorial and binomial topics
Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is given by: i.e., the sum of the first k nth-order binomial coefficients. The first rows of Bernoulli's triangle are: Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if denotes the component in row n and column k, then: As in Pascal's triangle and other similarly constructed triangles, sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers. As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2. The fourth column (k = 3) is the three-dimensional analogue, known as the cake numbers, for n cuts, where n ≥ 3. The fifth column (k = 4) gives the maximum number of regions in the problem of dividing a circle into areas for n + 1 points, where n ≥ 4. In general, the (k + 1)th column gives the maximum number of regions in k-dimensional space formed by n − 1 (k − 1)-dimensional hyperplanes, for n ≥ k. It also gives the number of compositions (ordered partitions) of n + 1 into k + 1 or fewer parts. (Wikipedia).
What are the trigonometric definitions of a right triangle
👉 Learn all about the trigonometry of right triangles. A right triangle is a triangle that has 90 degrees as one of its angles. The trigonometric identities of right triangles give us the relationship between the angles of a right triangle and the side lengths of the right triangle. These
From playlist Right Triangle Trigonometry | Learn About
Using the Isosceles triangle theorem to find the measure of x
👉 Learn how to find the missing side of a triangle. A triangle is a polygon with three sides. Triangles are classified on the basis of the angles or on the basis of the sides. The classification of a triangle on the basis of the sides are: scalene, isosceles, and equilateral triangles. A
From playlist Triangles
Label the parts of a triangle ex 1
👉 Learn how to find the missing side of a triangle. A triangle is a polygon with three sides. Triangles are classified on the basis of the angles or on the basis of the sides. The classification of a triangle on the basis of the sides are: scalene, isosceles, and equilateral triangles. A
From playlist Triangles
Billiard in a Reuleaux triangle
Seeing the billiard in a Reuleaux triangle was a wish by several viewers (including, I believe, Carmen, Bogdan, Auferen, Bluelightzero and Jonathan), so here it finally is! The Reuleaux triangle is a shape of constant width. You can build it by starting from an equilateral triangle, and t
From playlist Particles in billiards
What are the parts of a right triangle
👉 Learn all about the trigonometry of right triangles. A right triangle is a triangle that has 90 degrees as one of its angles. The trigonometric identities of right triangles give us the relationship between the angles of a right triangle and the side lengths of the right triangle. These
From playlist Right Triangle Trigonometry | Learn About
This is a recreation of a short clip from a long form video showing six different ways to construct the Sierpinski triangle: https://youtu.be/IZHiBJGcrqI In this short, we shade odd entries of the Halayuda/Pascal triangle to obtain the Sierpinski triangle. Can you explain why this works?
From playlist Fractals
👉 Learn the essential definitions of triangles. A triangle is a polygon with three sides. Triangles are classified on the basis of their angles or on the basis of their side lengths. The classification of triangles on the bases of their angles are: acute, right and obtuse triangles. The cl
From playlist Types of Triangles and Their Properties
Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)
The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :) Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and
From playlist Recent videos
How do you find all of the sides for a equilateral triangle
👉 Learn how to find the missing side of a triangle. A triangle is a polygon with three sides. Triangles are classified on the basis of the angles or on the basis of the sides. The classification of a triangle on the basis of the sides are: scalene, isosceles, and equilateral triangles. A
From playlist Triangles
Advice for research mathematicians | Bernoulli numbers and Faulhaber's sums of powers | WIld Egg
The Bernoulli numbers are an intriguing family of rational numbers that arise in many areas of analysis. We introduce them in the context of J. Faulhaber's formulas for sums of powers of natural numbers, which in fact give us another important family of polynomials or polynumbers. These n
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Statistical analysis of networks - Professor Gesine Reinert, University of Oxford
Networks have become increasingly popular as representations of complex data. How can we make sense of such data? The first class will cover some network summaries and some parametric models for networks, while the second class concerns statistical inference using these network summaries a
From playlist Data science classes
Advice | Variants of the Bernoulli numbers via an AI approach to maths research | Wild Egg Maths
We advocate a simple minded AI approach to pure maths research: start with a basic, central object in mathematics, and just systematically explore the possibilities adjacent to it, where adjacent means roughly that we perform small variants and see what happens. To illustrate the strategy
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Lecture 4 - Perceptron & Generalized Linear Model | Stanford CS229: Machine Learning (Autumn 2018)
For more information about Stanford’s Artificial Intelligence professional and graduate programs, visit: https://stanford.io/3GnSw3o Anand Avati PhD Candidate and CS229 Head TA To follow along with the course schedule and syllabus, visit: http://cs229.stanford.edu/syllabus-autumn2018.h
From playlist Stanford CS229: Machine Learning Full Course taught by Andrew Ng | Autumn 2018
The Basel Problem Part 1: Euler-Maclaurin Approximation
This is the first video in a two part series explaining how Euler discovered that the sum of the reciprocals of the square numbers is π^2/6, leading him to define the zeta function, and how Riemann discovered the surprising connection between the zeroes of the zeta function and the distrib
From playlist Analytic Number Theory
The Brachistochrone, with Steven Strogatz
Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos
From playlist 3Blue1Brown | Math for fun and glory | Khan Academy
Continuous Distributions: Beta and Dirichlet Distributions
Video Lecture from the course INST 414: Advanced Data Science at UMD's iSchool. Full course information here: http://www.umiacs.umd.edu/~jbg/teaching/INST_414/
From playlist Advanced Data Science
Statistical inference for networks: Professor Gesine Reinert, University of Oxford
Professor Gesine Reinert, Oxford University Research interests Applied Probability, Computational Biology, and Statistics. In particular: Stein’s method, networks, word count statistics Have you heard about the phenomenon that everyone is six handshakes away from the President? The six d
From playlist Data science classes
What is the converse of the Pythagorean Theorem
👉 Learn all about the trigonometry of right triangles. A right triangle is a triangle that has 90 degrees as one of its angles. The trigonometric identities of right triangles give us the relationship between the angles of a right triangle and the side lengths of the right triangle. These
From playlist Right Triangle Trigonometry | Learn About
Faulhaber's Formula and Bernoulli Numbers | Algebraic Calculus One | Wild Egg
This is a lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra, and avoiding all "infinite processes", "real numbers" and other modern fantasies. The course will be carefully framed on
From playlist Algebraic Calculus One from Wild Egg
Learn to write the three trigonometric function ratios from a triangle
👉 Learn all about the trigonometry of right triangles. A right triangle is a triangle that has 90 degrees as one of its angles. The trigonometric identities of right triangles give us the relationship between the angles of a right triangle and the side lengths of the right triangle. These
From playlist Right Triangle Trigonometry | Learn About