f_2(x) = (1 + x)(1 + x/2) =(x+1)(x+2)/2!
f_4(x) = (1 + x)(1 + x/2)(1 + x/3)(1 + x/4) =(x+1)(x+2)(x+3)(x+4)/4!
f_6(x) = (1 + x)(1 + x/2)(1 + x/3)(1 + x/4)(1 + x/5)(1 + x/6) =(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)/6!
f_8(x) = (1 + x)(1 + x/2)(1 + x/3)(1 + x/4)(1 + x/5)(1 + x/6)(1 + x/7)(1 + x/8) = (x+1)(x+2)(x+3)(x+4)(x+5)(x+6)(x+7)(x+8)/8!
and in general :
f_2n(x) = (x + 2n)! / [x! (2n)!]
So related is the gamma function.
We have the following pattern for n > 0 :
f_2n(x) = g(x^2 + (2n+1) x)
where g is a polynomial with rational coefficients of degree n.
Examples :
f_8(x) = (x^2 + 9x)^4 / 8! + 1/672 (x^2 + 9x)^3 + 109/3360 (x^2 + 9x)^2 + 761/2520 (x^2 + 9x) + 1
f_6(x) = 1/6! (x^2 + 7x)^3 + 7/180(x^2 + 7x)^2 + 7/20 (x^2 + 7x) + 1
f_4(x) = 1/4! (x^2 + 5x)^2 + 5/12 (x^2 + 5x) + 1
Prove it.