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Binomial coefficient

The binomial coefficients for the terms of in the expansion of the binomial theorem with negative exponents, can be written as

The upper number n is the exponent from the binomial theorem, the lower number k is the consecutive number of the term in the outcome.

 

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Explanation

A number arises from the sum of the numbers right above it and left above it. You see it at −120 = −330 + 210.

n		0	1	2	3	4	5	6	7	8	k → ∞
···		···	···	···	···	···	···	···	···	···	···
−5		1	−5	15	−35	70	−126	210	−330	495	···
−4		1	−4	10	−20	35	−56	84	−120	165	···
−3		1	−3	6	−10	15	−21	28	−36	45	···
−2		1	−2	3	−4	5	−6	7	−8	9	···
−1		1	−1	1	−1	1	−1	1	−1	1	···
0		1	◦	◦	◦	◦	◦	◦	◦	◦	···
1		1	1	◦	◦	◦	◦	◦	◦	◦	···
2		1	2	1	◦	◦	◦	◦	◦	◦	···
3		1	3	3	1	◦	◦	◦	◦	◦	···
4		1	4	6	4	1	◦	◦	◦	◦	···
5		1	5	10	10	5	1	◦	◦	◦	···
6		1	6	15	20	15	6	1	◦	◦	···
7		1	7	21	35	35	21	7	1	◦	···
8		1	8	28	56	70	56	28	8	1	···
···		···	···	···	···	···	···	···	···	···	···

It is immediately noticeable that for the negative exponents every row has an infinite number of terms and that alternating positive and negative coefficients arise, which are getting bigger and bigger. For negative exponents the formula is

This formula cannot be used for positive exponents. We calculate the binomial coefficients of (a + b)⁻⁴ and find

 

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