(48G/50g) Binomial Transform, Difference Table
01-18-2019, 08:50 PM
Post: #6
 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013
RE: (48G/50g) Binomial Transform, Difference Table
That's just this formula for the inverse binomial transform from the linked Wikipedia article:

$$a_n=\sum_{k=0}^n {n\choose k} t_k$$

In the given example it leads to:

\begin{align*} a_n&=0\binom{n}{0}+1\binom{n}{1}+2\binom{n}{2}+1\binom{n}{3}\\ &= 0+1\frac{n}{1}+2\frac{n(n-1)}{2}+1\frac{n(n-1)(n-2)}{6}\\ &=n+ n(n-1)+\frac{n(n-1)(n-2)}{6}\\ &=\frac{n(n+1)(n+2)}{6} \end{align*}

This is in accordance with A000292.

Cheers
Thomas
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 Messages In This Thread (48G/50g) Binomial Transform, Difference Table - John Keith - 01-17-2019, 09:56 PM RE: (48G/50g) Binomial Transform, Difference Table - Thomas Klemm - 01-18-2019, 05:53 AM RE: (48G/50g) Binomial Transform, Difference Table - John Keith - 01-18-2019, 08:01 PM RE: (48G/50g) Binomial Transform, Difference Table - Thomas Klemm - 01-18-2019, 08:14 AM RE: (48G/50g) Binomial Transform, Difference Table - John Keith - 01-18-2019, 07:39 PM RE: (48G/50g) Binomial Transform, Difference Table - Thomas Klemm - 01-18-2019 08:50 PM RE: (48G/50g) Binomial Transform, Difference Table - John Keith - 01-18-2019, 09:44 PM RE: (48G/50g) Binomial Transform, Difference Table - John Keith - 05-27-2019, 09:07 PM

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