求{∞a,∞b,∞c,∞d}的n-组合,其中a出现偶数次,b出现奇数次,要求用指数生成函数求解我求出来是4的n-1次方,但明显不对.求指教

Question

求{∞a,∞b,∞c,∞d}的n-组合,其中a出现偶数次,b出现奇数次,要求用指数生成函数求解
我求出来是4的n-1次方,但明显不对.求指教

Answer 1

It will be convenient to deal with by using ordinary generating functions,i.e.,
x/(1-x^2)^2/(1-x)^2.
Then find the explicit formula by checking OEIS A006918.Such a formula is
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48.
To use exponential generating functions,you need a good explanation for the relation between what you want and the coefficients of the e.g.f.