Pascal's Triangle and Its Relatives

Background

Binomial Expansion

The common explanation for our interest in Pascal's triangle is based on the expansion of the binomial (x + y)n:

(x + y)0 = 1 (x + y)1 = x + y (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2y + 3xy2 + y3 (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4 ...

Compare these coefficients with the entries in Pascal's triangle.

The relation between these coefficients and the Pascal triangle construction is easy to understand by induction.

For example, write (x + y)n = anxn + an-1xn-1y + an-2xn-2y2 + ... + a1xyn-1 + a0yn

and (x + y)n+1 = bn+1xn+1 + bnxny + bn-1xn-1y2 + ... + b1xyn + b0yn+1

where an = a0 = bn+1 = b0 = 1.

Noting that (x + y)n+1 = (x + y)n(x + y), to deduce the bi from the ai, we multiply (x + y)n by x and by y, then add the results.

anxn + an-1xn-1y + an-2xn-2y2 + ... + a1xyn-1 + a0yn x + y anxn+1 + an-1xny + an-2xn-1y2 + ... + a1x2yn-1 + a0xyn anxny + an-1xn-1y2 + ... + a2x2yn-1 + a1xyn + a0yn+1 anxn+1 + (an+an-1)xny + (an-1+an-2)xn-1y2 + ... + (a2+a1)x2yn-1 + (a1+a0)xyn + a0yn+1

So we see bn+1 = an = 1, b0 = a0 = 1, and for all i, n <= i <= 1, bi = ai + ai-1.

Noting how the rows grow to the right, this is exactly the Pascal's triangle relation.

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