TITLE: What are some easy but beautiful patterns in Pascal's Triangle?
QUESTION [17 upvotes]: Pascal's triangle has wide applications in Mathematics.I have seen that the most important applications relate to the binomial coefficients and combinatorics.
Are there some other beautiful interesting patterns in Pascal's triangle that can be found by selecting some other combination modes ?

REPLY [8 votes]: Here's a way to extend Pascal's triangle; I think it's interesting.
In black we have the original Pascal's triangle. In green we have inserted rows between the black rows. To find the values, we use the polynomials which describe the slanted columns of the original triangle, shown in blue, and evaluate at half integers.

So what's the point of this? Remember the numbers in Pascal's triangle also provide the binomial coefficients.
For example, $(1+x)^4=1+4x+6x^2+4x^3+x^4$.
These new rows also describe the coefficients of the series for binomials raised to the half integer power:
$(1+x)^\frac{7}{2}=1+\frac{7}{2}x+\frac{35}{8}x^2+\frac{35}{16}x^3+...$
Unlike the rows of Pascal's triangle, these continue forever. Also, like Pascal's tringle, the green numbers are the sum of the two green numbers directly above. For example, $\frac{15}{8}+\frac{5}{16}=\frac{35}{16}$.