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
, as
- if or , and otherwise
-
In the sequel we will look only at the case . For this case we will
introduce a more persuasive notation and a different enumeration. We set
-
This gives the following symmetric representation
-
We see that the central term of this generalization of the binomial
coefficients is the swinging factorial. This means
-
We see directly that
-
Also the symmetry relation is clear
-
If we set
and
,
then the generalized binomial coefficients can be computed for
and
by the recurrence
-
To show this we introduce the abbreviation
-
Now the recurrence equation can be written
-
If and this is equivalent to
-
The left hand side of the equation is , and the right hand side also, because
-
-
-
Looking at table 1 below we see the classical binomial triangle embedded:
the pictorial representation of Pascal's triangle
originates from
by deleting those entries, where and have a different parity.
On formal ground we have
-
Appendix
Tabel 1.
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.
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 |
|
A056040 |
|
A098361 |
|
A180274 |
|
A180064 |
|
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: