SOLUTION: What is the recursive relation or generating function for the following: (a) 4, 12, 24, 24, 12, -12, -48, -96, -156, -228, .... (b) 4, 12, 24, 24, 12, 4, 1, 1/5, 1/30, 1/210, 1

Question 1163947: What is the recursive relation or generating function for the following:
(a) 4, 12, 24, 24, 12, -12, -48, -96, -156, -228, ....
(b) 4, 12, 24, 24, 12, 4, 1, 1/5, 1/30, 1/210, 1/1680, ....
(c) 1, 5, 20, 60, 120, 120, 0, 0, 0, ....
(d) 1, 5, 20, 60, 120, 120, 60, 20, 5, 1, 1/6, 1/42, 1/336, ....
Answer by CPhill(2264) (Show Source): You can put this solution on YOUR website!
Finding the recursive relations and generating functions for these sequences involves identifying the pattern of multiplication or the relationship between the coefficients of a polynomial (like the binomial expansion).
### (a) 4, 12, 24, 24, 12, -12, -48, -96, -156, -228, ...
This sequence is based on the **third differences** being constant (eventually). By calculating the differences between terms:
* **1st Differences:** 8, 12, 0, -12, -24, -36, -48, -60, -72...
* **2nd Differences:** 4, -12, -12, -12, -12, -12, -12...
* **3rd Differences:** -16, 0, 0, 0, 0, 0...
**Recursive Relation:**
After $n=3$, the second difference is constant ($-12$). The general recursive form for a sequence with constant second differences is:
$$a_n = 3a_{n-1} - 3a_{n-2} + a_{n-3}$$
*(Note: This holds true for the later part of the sequence where the pattern stabilizes.)*
**Generating Function:**
Since it eventually follows a quadratic pattern:
$$G(x) = \frac{4 + x - 8x^2 - 3x^3}{(1-x)^3}$$
---
### (b) 4, 12, 24, 24, 12, 4, 1, 1/5, 1/30, 1/210, ...
This sequence is formed by dividing by an increasing integer at each step, related to **$\frac{4! \binom{n}{k}}{n!}$** or variations of **$\frac{1}{n!}$**. Specifically, starting from the 4th term:
* $24 \div 1 = 24$
* $24 \div 2 = 12$
* $12 \div 3 = 4$
* $4 \div 4 = 1$
* $1 \div 5 = 1/5$
**Recursive Relation:**
$$a_n = a_{n-1} \cdot \frac{k}{n} \text{ (where } k \text{ is a constant index shift)}$$
More simply: $a_n = \frac{4 \cdot 4!}{(n-1)!}$ for $n \ge 4$ (with manual overrides for the start).
**Generating Function:**
This is a variation of the exponential generating function $e^x$:
$$G(x) = 4 + 12x + 24x^2 + 24x^3 + 12x^4 + 4x^5 + \sum_{n=6}^{\infty} a_n x^n$$
---
### (c) 1, 5, 20, 60, 120, 120, 0, 0, 0, ...
These are the coefficients of the polynomial for **$5! \binom{5}{k} / k!$** or simply the values of **$P(5, k)$** (permutations).
* $P(5,0) = 1$
* $P(5,1) = 5$
* $P(5,2) = 20$
* $P(5,3) = 60$
* $P(5,4) = 120$
* $P(5,5) = 120$
* $P(5,6) = 0$
**Recursive Relation:**
$$a_n = a_{n-1} \cdot (6-n) \text{ for } n \le 6, \text{ else } 0$$
**Generating Function:**
This is a finite polynomial:
$$G(x) = 1 + 5x + 20x^2 + 60x^3 + 120x^4 + 120x^5$$
---
### (d) 1, 5, 20, 60, 120, 120, 60, 20, 5, 1, 1/6, 1/42, ...
This is a **symmetric** version of sequence (c). It represents the coefficients of the expansion of a binomial power where the terms are divided by factorials, often seen in **Hypergeometric series**.
**Recursive Relation:**
$$a_n = a_{n-1} \cdot \frac{6-n}{n}$$
*(Notice: at $n=1$, $5/1=5$; at $n=2$, $4/2=2 \to 20$; etc. Once $n > 5$, the terms become fractional.)*
**Generating Function:**
This is the generating function for the binomial coefficients $\binom{5}{n}$:
$$G(x) = (1+x)^5$$
*(However, because of the fractional tail $1/6, 1/42$, this specifically matches the coefficients of the expansion of a **generalized binomial series** or a **Hypergeometric function** $_1F_0$.)*
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