# Is there a closed form expression for the power series whose coefficients are half of the binomial series?

Quite similarly to this question, given the following expression: $$P_n(x) = \sum_{0 \leq k \leq n/2} \binom{n}{k} x^k$$ Is it possible to find a closed form expression for: $$G(x,y) = \sum_{n=0}^{+\infty} P_n(x) \frac{y^n}{n!}$$

(The difference with the former question is that here, there is an additional $$k!$$ in the expression of $$G(x,y)$$)

A few things that I seem to notice:

• for $$x=0$$, we obviously have: $$G(0,y) = e^y$$
• for $$x$$ small and $$y$$ large, it seems numerically quite close to $$e^{(1+x)y}$$

If you have any idea if this is possible to find another expression without involving power series for $$G(x,y)$$ I would be very grateful! Thank you !