Binomial Coefficients and Generating Functions - Cs.ioc.ee

Sequence m
0
Let’s take sequences
and
,0,− m
1
,0, m
2
,0,− m
3
,0, m
4
m m m m
〈 , , ,..., ,...〉 ←→ F (X ) = (1 + x)
0 1 2 n
m
,0,...,
m m m m
〈 ,− , ,− ,...,(−1)
0 1 2 3
n
m
,...〉 ←→ G(X ) = (1 − x)
n
m
Then the convolution corresponds to the function (1 − x) m (1 + x) m = (1 − x 2 ) m that
gives the identity of binomial coefficients:
n
m m
∑
(−1)
j=0 j n − j
j = (−1) n/2
m
[n is even].
n/2