What is the Formula for the Numerator in Factorial Series Calculations?

[m_state724]

Ok, so here goes nothing

I have predict what this is in series form, the factorial in the numerator is really throwing me off. I only have to do the series for the first six terms.

C0=C0
C2= -5/2 C0
C4= -3/4 C2 = (5x3)/(2x4) C0
C6= -1/6 C4 = -(5x3x1)/(2x4x6) C0

This is what I have so far, although I cannot completely find out the numerator:

n is 0,1,2,3
C(2n)=[(-1)^n][?]/[(2n)!] C0
or
C(2n)=[(-1)^n][?]/[2^n x n!] C0

I cannot find out how to correctly represent the numerator. Can you guys give me any pointers?

 

[driscoll79]

I may be missing something here, but this doesn't look like a factorial to me. It looks like $$\prod_{k=1}^4 \frac{2k-1}{2k}$$. does it have to be in factorial form?