Jonathan. Frech’s WebBlog

Pascal’s Triangle (#111)

Jonathan Frech

Pascal’s triangle is an interesting mathematical sequence. It is often written as a triangle, starting with $\{1\}$$\{1\}$, then $\{1,1\}$$\{1,1\}$. To gen­er­ate the next row, you add the two numbers above to form another one. So the next row in the sequence is $\{1,2,1\}$$\{1,2,1\}$ then $\{1,3,3,1\}$$\{1,3,3,1\}$, $\{1,4,6,4,1\}$$\{1,4,6,4,1\}$ and so on (sequence A007318 in OEIS).

One interesting property of Pascal’s triangle is the generation of binomials.
To calculate $(a+b)^4$$(a+b)^4$, you can look at the 4-th row (listed above and when starting to count at 0) and de­ter­mine

$$(a+b)^4=(1\cdot a^4\cdot b^0)+(4\cdot a^3\cdot b^1)+(6\cdot a^2\cdot b^2)+(4\cdot a^1\cdot b^3)+(1\cdot a^0\cdot b^4)$$$$(a+b)^4=(1\cdot a^4\cdot b^0)+(4\cdot a^3\cdot b^1)+(6\cdot a^2\cdot b^2)+(4\cdot a^1\cdot b^3)+(1\cdot a^0\cdot b^4)$$
$$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4.$$$$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4.$$

This program generates Pascal’s sequence in a rather unusual shape, looking a bit like a crown.

To get more information about Pascal’s triangle, check out this Wikipedia entry.

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Source code: pascals-triangle.py

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[1]In pascals-triangle_pascal.py there is a typo: “sedonds” should say “seconds”.