Pascal's triangle has a lot of mathematically interesting properties; It represents binomial coefficients (n choose k or combination of a set), you can find within it the powers of 2, the powers of 11, the Fibonacci sequence, Sierpinski's triangle (a fractal), all the figurate numbers, Mersenne numbers and Catalan numbers, just to name a few. Furthermore, generating Pascal's triangle is quite simple, requiring only addition. Sometimes in mathematics something is both profound AND simple to understand. Such is the case with Pascal's triangle.

Recently, I made a contribution to rosettacode.org. If you have not checked out rosettacode.org before, you should definitely do so. I made a contribution to the Pascal's triangle task for the C# language. You can check it out here, or you can just view the code below.

My version of Pascal's Triangle is short, succinct, uses the BigInteger class for arbitrarily large numbers, and uses the algorithm to generate a single row of Pascal's triangle without needing to generate every row before it. This was originally the use case for writing this code in the first place; I wanted to generate high-numbered rows in a computationally feasible way. What does row # 5000 look like, for example? Well, could use the code below and generate the 5000th row of Pascal's triangle and format it as a string with the following one-liner: `string result = string.Join(" ", PascalsTriangle.GetRow(5000).Select(n => n.ToString()));`. You can also, of course, generate actual triangle-shaped rows of numbers:

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 1 12 66 220 495 792 924 792 495 220 66 12 1 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1 1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1 1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1 1 19 171 969 3876 11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876 969 171 19 1

Awww, yes... that's pleasing.

And the code to do all that is thus:

```
public static class PascalsTriangle
{
public static IEnumerable
``` GetTriangle(int quantityOfRows)
{
IEnumerable range = Enumerable.Range(0, quantityOfRows).Select(num => new BigInteger(num));
return range.Select(num => GetRow(num).ToArray());
}
public static IEnumerable GetRow(BigInteger rowNumber)
{
BigInteger denominator = 1;
BigInteger numerator = rowNumber;
BigInteger currentValue = 1;
for (BigInteger counter = 0; counter <= rowNumber; counter++)
{
yield return currentValue;
currentValue = BigInteger.Multiply(currentValue, numerator--);
currentValue = BigInteger.Divide(currentValue, denominator++);
}
yield break;
}
public static string FormatTriangleString(IEnumerable triangle)
{
int maxDigitWidth = triangle.Last().Max().ToString().Length;
IEnumerable rows = triangle.Select(arr =>
string.Join(" ", arr.Select(array => CenterString(array.ToString(), maxDigitWidth)) )
);
int maxRowWidth = rows.Last().Length;
return string.Join(Environment.NewLine, rows.Select(row => CenterString(row, maxRowWidth)));
}
private static string CenterString(string text, int width)
{
int spaces = width - text.Length;
int padLeft = (spaces / 2) + text.Length;
return text.PadLeft(padLeft).PadRight(width);
}
}

Note: This requires the System.Numerics library.I make liberal use of Linq to keep the code short, yet expressive.

The code that generated the Pascal's triangle above, is:

```
IEnumerable
``` triangle = PascalsTriangle.GetTriangle(20);
string output = PascalsTriangle.FormatTriangleString(triangle)
Console.WriteLine(output);

Other, arbitrarily large, number types available include the BigDecimal class, the BigComplex class, and the BigRational class. They are all available on my GitHub.

Thanks for stopping by!