Concrete mathematics : a foundation for computer science

174 BINOMIAL COEFFICIENTS
Table 174 The ton ten binomial coefficient identities.
n
0k
=-n!
k!(n--k)! ’
integers
nak>O.
(E) = (n.l.k) ’
integer n 3 0,
integer k.
factorial expansion
symmetry
integer k # 0. absorption/extraction
(;) = (Ii’) + (;I:), integer k. addition/induction
(;) = (-l)k(kVL-‘), integer k. upper negation
integers m, k. trinomial revision
integer r 3 0,
or Ix/y1 < 1.
binomial theorem
integer n. parallel summation
integers
m,n>O.
upper summation
integer n. Vandermonde convolution
And now we can apply the parallel summation identity, (5.9):
n-mfk ‘(n-m) +m+ 1
k m
\
) = (n;‘).
Finally’ we reinstate the (k) in the denominator that we removed from
the sum earlier, and then apply (5.7) to get the desired closed form:
(“;‘)/(:) = $A&*
This derivation actually works for any real value of n, as long as no division
by zero occurs; that is, as long as n isn’t one of the integers 0, 1, . . . , m - 1.