Binomial Coefficients and Generating Functions - Cs.ioc.ee

More examples (3)
Taking in the last example a = 0,5 and b = 1 yields
0,5 + 0,5x + 0,5x 2 + 0,5x 3 + ··· = 0,5
1 − x
Taking a = 0,5 and b = −1, gives
0,5 − 0,5x + 0,5x 2 − 0,5x 3 + ··· = 0,5
1 + x
Adding equations (1) and (2), we get the generating function of the
sequence 〈1,0,1,0,1,0,...〉:
1 + x 2 + x 4 + x 6 + ··· = 0,5 0,5
+
1 − x 1 + x =
1
1
=
(1 − x)(1 + x) 1 − x 2
(1)
(2)