https://github.com/kolosovpetro/identitiesinrascaltriangle

In this manuscript, we show new binomial identities in iterated rascal triangles, revealing a connection between the Vandermonde convolution and iterated rascal numbers. We also present Vandermonde-like binomial identities. Furthermore, we establish a relation between iterated rascal triangle and (1,q)-binomial coefficients.
https://github.com/kolosovpetro/identitiesinrascaltriangle

binomial-coefficient binomial-theorem mathematica mathematics pascals-triangle

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In this manuscript, we show new binomial identities in iterated rascal triangles, revealing a connection between the Vandermonde convolution and iterated rascal numbers. We also present Vandermonde-like binomial identities. Furthermore, we establish a relation between iterated rascal triangle and (1,q)-binomial coefficients.

• Host: GitHub
• URL: https://github.com/kolosovpetro/identitiesinrascaltriangle
• Owner: kolosovpetro
• License: gpl-3.0
• Created: 2024-06-27T16:04:15.000Z (4 months ago)
• Default Branch: main
• Last Pushed: 2024-09-14T14:31:23.000Z (about 2 months ago)
• Last Synced: 2024-09-15T00:19:44.133Z (about 2 months ago)
• Topics: binomial-coefficient, binomial-theorem, mathematica, mathematics, pascals-triangle
• Language: TeX
• Homepage: https://kolosovpetro.github.io/pdf/IdentitiesInRascalTriangle.pdf
• Size: 5.4 MB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Changelog: CHANGELOG.md
  • License: LICENSE

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README

        
# Identities in Iterated Rascal Triangles

In this manuscript, we introduce new binomial identities in iterated Rascal triangles,

uncovering a connection between Vandermonde convolution and iterated Rascal numbers.

Additionally, we present novel identities involving the finite differences of iterated Rascal numbers

and binomial coefficients.

The manuscript also offers a proof of the row sums conjecture for iterated Rascal triangles.

Furthermore, we establish and explore the relationship between iterated Rascal triangles

and $(1,q)$-binomial coefficients, highlighting connections to relevant OEIS sequences.

All results are supported by supplementary Mathematica programs for validation.

## Build and run in Intellij IDEA

- Install `MikTeX`: https://miktex.org/download

- Update `MikTeX`

- Install `SumatraPDF` viewer: https://www.sumatrapdfreader.org/download-free-pdf-viewer

- Path to SumatraPDF: `C:\Program Files\SumatraPDF`

- Install `Intellij IDEA Ultimate`: https://www.jetbrains.com/idea/download/#section=windows

- Activate `Intellij IDEA Ultimate`

- Install `TeXiFy IDEA` plugin: https://plugins.jetbrains.com/plugin/9473-texify-idea

- Clone this repository locally: `https://github.com/kolosovpetro/github-latex-template.git`

- Open `github-latex-template` folder in `Intellij IDEA Ultimate` and configure as follows

    - LaTeX Configuration

      ![LaTeX Configuration](./img/latex_configuration.PNG "LaTeX Configuration")

    - BibTeX Configuration

      ![BibTeX Configuration](./img/bibtex_configuration.PNG "BibTeX Configuration")

- Configure Inverse Search in `Intellij IDEA` for SumatraPDF: `Tools -> LaTeX -> Configure Inverse Search`