Generalized Binomial   dsc-8.1

Peter Luschny

Our goal is to define a triangle of binomial coefficients where the swinging factorial is always the middle coefficient, and not only in the case when $n$ is even .

To this end we define the generalized binomial coefficient for integer $n \geq 0$, $m \geq 1$ as $\operatorname{C}_{n,k}^{(m)} = 0$ if $k<0$ or $k>mn$, and otherwise

$$\operatorname{C}_{n,k}^{(m)}=\frac{n!} {(n-\left\lceil k/m\right\rceil)!\left\lfloor k/m\right\rfloor !} \qquad\left(0\leq k\leq mn\right) \, .$$

In the sequel we will look only at the case $m=2$. For this case we will introduce a more persuasive notation and a different enumeration. We set

$$\binom{n}{k}_{2} = \operatorname{C}_{n,\,n+k}^{(2)}\qquad \left(n\geq0,\ -n\leq k\leq n\right) \ .$$

This gives the following symmetric representation

$$\binom{n}{k}_{2} = \frac{n!}{{\Omega_{n-k}! \ \Omega_{n+k}!}} \ , \quad \text{where} \ \Omega_{x} := \frac{x - \isodd{x}}{2} \ . \label{symbin}$$

We see that the central term of this generalized binomial coefficient is the swinging factorial. This means

$$\binom{n}{0}_{2}= \swing{n}\, .$$

We see directly that

$$\binom{n}{n}_{2} = 1, \ \binom{n}{n-1}_{2} = n .$$

Also the symmetry relation is clear

$$\binom{n}{k}_{2}= \binom{n}{-k}_{2} \quad\left(0\leq k\leq n\right) \ .$$

If we set $\binom{n}{n}_{2}=\binom{n}{-n}_{2}=1$ and $\binom{n}{n-1}_{2} = \binom{n}{1-n}_{2} = n$, then the generalized binomial coefficients can be computed for $n \geq 2$ and $-n+2 \leq k \leq n-2$ by the recurrence

$$\binom{n}{k}_{2} = \binom{n-1}{k-1}_{2}+ \isodd{n-k} \binom{n-1}{k}_{2}+\binom{n-1}{k+1}_{2} . \label{binrec}$$

To show this we introduce the abbreviation

$$s_{n,k}=\Omega_{n-k}! \ \Omega_{n+k}! \ .$$

Now the recurrence equation can be written

$$\frac{n!}{s_{n,k}}=\frac{(n-1)!}{s_{n-1,k-1}}+ \isodd{n-k}\frac{(n-1)!}{s_{n-1,k}}+\frac{(n-1)!}{s_{n-1,k+1}}\ .$$

If $n \geq 2$ and $-n+2 \leq k \leq n-2$ this is equivalent to

$$\frac{n!}{\left(n-1\right)!}=\frac{s_{n,k}}{s_{n-1,k-1}}+ \isodd{n-k}\frac{s_{n,k}}{s_{n-1,k}}+\frac{s_{n,k}}{s_{n-1,k+1}} \ .$$

The left hand side of the equation is $n$, and the right hand side also, because

$$\begin{align*} \frac{s_{n,k}}{s_{n-1,k-1}} & =\frac n2+\frac k2- \frac12 \isodd{n-k} \ ,\\ \isodd{n-k} \frac{s_{n,k}}{s_{n-1,k}} & =\isodd{n-k} \ ,\\ \frac{s_{n,k}}{s_{n-1,k+1}} & =\frac n2-\frac k2- \frac12\isodd{n-k} \,. \quad\diamond \end{align*}$$

Looking at table [1] we see the classical binomial triangle embedded: the pictorial representation of Pascal's triangle $\binom nk$ originates from $\binom nk_{2}$ by deleting those entries, where $n$ and $k$ have a different parity. On formal ground we have

$$\binom{n}{k} = \binom{n}{2k-n}_{2} \qquad \left(0 \leq k \leq n \right) \ .$$

Appendix

Tabel 1. $\ \binom{n}{k}_{2} \ $

n \ k	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
0							1
1						1	1	1
2					1	2	2	2	1
3				1	3	3	6	3	3	1
4			1	4	4	12	6	12	4	4	1
5		1	5	5	20	10	30	10	20	5	5	1
6	1	6	6	30	15	60	20	60	15	30	6	6	1

Tafel 2. $ \ \ \Omega_{n-k}! \ \Omega_{n+k}! \ = \ n! \ / \ \binom{n}{k}_{2} $

n \ k	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6
0							1
1						1	1	1
2					2	1	1	1	2
3				6	2	2	1	2	2	6
4			24	6	6	2	4	2	6	6	24
5		120	24	24	6	12	4	12	6	24	24	120
6	720	120	120	24	48	12	36	12	48	24	120	120	720

Sequences

OEIS

A162246	$\ \operatorname{C}_{n,k}^{(2)} \ $
A056040	$ \swing{n}$
A098361	$n! (n-k)!$
A180274	$ \Omega_{n-k}! \ \Omega_{n+k}! $
A180064	$n! \ / \swing{n}$

Code-Snippets

C := proc(m,n,k)
if k < 0 or k > m*n then 0 else
n!/(floor(k/m)!*(n-ceil(k/m))!) fi end:
C2 := proc(n,k) C(2,n,n+k) end;
Omega := proc(n) (n-(n mod 2))/2 end: 
C2o := proc(n,k) n!/(Omega(n-k)!*Omega(n+k)!) end:

Links

Rational Trees	Factorial Functions	Eulerian Polynomials	vonStaudt Primes	Irregular Primes	Generalized Binomial
Bernoulli Confusion	Partitions Stirling	Zeta Polynomials	Perfect Rulers	Swinging Primes	Clausen Numbers
Hadamard Gamma	History Factorial	Permutation Trees	Swiss Knife	Swinging Factorial	Worpitzky Triangle