ABSTRACT
Definitions (special cases): Ckn =
(n k
) =
n! k! (n – k)! , where k = 1, . . . ,n;
C0a = 1, Cka = (a k
) = (–1)k (–a)k
k! =
a(a – 1) . . . (a – k + 1) k!
, where k = 1, 2, . . .
Here a is an arbitrary real number. Definition (general case):
Cba = Γ(a + 1)
Γ(b + 1)Γ(a – b + 1) , where Γ(x) is the gamma function. Properties:
C0a = 1, Ckn = 0 for k = –1, –2, . . . or k > n,
Cb+1a = a
b + 1 Cba-1 =
a – b
b + 1 Cba, C
b a + C
b+1 a = C
Cn –1/2 =
(–1)n 22n
Cn2n = (–1)n (2n – 1)!! (2n)!! ,
Cn1/2 = (–1)n-1 n22n-1
Cn-12n-2 = (–1)n-1
n
(2n – 3)!! (2n – 2)!! ,
C2n+1n+1/2 = (–1)n2-4n-1Cn2n, Cn2n+1/2 = 2-2nC2n4n+1,
C1/2n = 22n+1
πCn2n , Cn/2n =
22n
π C(n-1)/2n ,
1 + C1n + C2n + · · · + Cnn = 2n, 1 – C1n + C2n – · · · + (–1)nCnn = 0.