User:Peter Luschny/GeneralizedBinomialCoefficients

Generalized Binomial Coefficients

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 $\scriptstyle n\geq 0$, $\scriptstyle m\geq 1$ as

${\text{C}}_{n,k}^{(m)}=0$ if $\scriptstyle k<0$ or $\scriptstyle k>mn$, and otherwise

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

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

${\binom {n}{k}}_{2}={\text{C}}_{n,\,n+k}^{(2)}\qquad \left(n\geq 0,\ -n\leq k\leq n\right)\ .$

This gives the following symmetric representation

${\binom {n}{k}}_{2}={\frac {n!}{\Omega _{n-k}!\ \Omega _{n+k}!}}\ ,\quad {\text{where}}\ \Omega _{x}:={\frac {x-\left[{x}\ {\text{odd}}\right]}{2}}\ .$

We see that the central term of this generalization of the binomial coefficients is the swinging factorial. This means

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

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

${\binom {n}{k}}_{2}={\binom {n-1}{k-1}}_{2}+\left[{n-k}\ {\text{odd}}\right]{\binom {n-1}{k}}_{2}+{\binom {n-1}{k+1}}_{2}\ .$

To show this we introduce the abbreviation

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

Now the recurrence equation can be written

${\frac {n!}{s_{n,k}}}={\frac {(n-1)!}{s_{n-1,k-1}}}+\left[{n-k}\ {\text{odd}}\right]{\frac {(n-1)!}{s_{n-1,k}}}+{\frac {(n-1)!}{s_{n-1,k+1}}}\ .$

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

${\frac {n!}{\left(n-1\right)!}}={\frac {s_{n,k}}{s_{n-1,k-1}}}+\left[{n-k}\ {\text{odd}}\right]{\frac {s_{n,k}}{s_{n-1,k}}}+{\frac {s_{n,k}}{s_{n-1,k+1}}}\ .$

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

${\frac {s_{n,k}}{s_{n-1,k-1}}}={\frac {n}{2}}+{\frac {k}{2}}-{\frac {1}{2}}\left[{n-k}\ {\text{odd}}\right]\ ,$

$\left[{n-k}\ {\text{odd}}\right]{\frac {s_{n,k}}{s_{n-1,k}}}=\left[{n-k}\ {\text{odd}}\right]\ ,$

${\frac {s_{n,k}}{s_{n-1,k+1}}}={\frac {n}{2}}-{\frac {k}{2}}-{\frac {1}{2}}\left[{n-k}\ {\text{odd}}\right].\quad \diamond$

Looking at table 1 below we see the classical binomial triangle embedded: the pictorial representation of Pascal's triangle $\scriptstyle {\binom {n}{k}}$ originates from $\scriptstyle {\binom {n}{k}}_{2}$ 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

Table 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	$\ {\text{C}}_{n,k}^{(2)}\$
A056040	${n}\wr$
A098361	$n!(n-k)!\quad$
A180274	$\Omega _{n-k}!\ \Omega _{n+k}!$
A180064	$n!\ /{n}\wr$

Maple

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: