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The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^{n-1} b_{p,n} x^{2p},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^p,$$ $$c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!},\;\;b_{p,n}=\frac{(-1)^{p+n+1}(n-1)!}{2 p!}\prod_{i=1}^{n-p}(4i-2).$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ was found by Peter Taylor.

The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^{n-1} b_{p,n} x^{2p},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^p,$$ $$b_{p,n}=\frac{(-1)^{p+n+1}(n-1)!}{2 p!}\prod_{i=1}^{n-p}(4i-2),\;\;c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!}.$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ was found by Peter Taylor.

The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^{n-1} b_{p,n} x^{2p},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^p,$$ $$c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!}.$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ still escapes me.

The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^{n-1} b_{p,n} x^{2p},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^p,$$ $$c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!},\;\;b_{p,n}=\frac{(-1)^{p+n+1}(n-1)!}{2 p!}\prod_{i=1}^{n-p}(4i-2).$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ was found by Peter Taylor.

The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^n b_{p,n} x^{2n},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^n,$$ $$c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!}.$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ still escapes me.

The general pattern is as follows: $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{{x}^{k}}{k!}\frac{1}{\prod_{i=1}^{n} (k+2i)}=\frac{1}{x^{2n}}\biggl(\frac{e^{-x}}{2^n n! }B_n(x)+C_n(x)\biggr),$$ $$B_n(x)=\sum_{p=0}^{n-1} b_{p,n} x^{2p},\;\;C_n(x)=\sum_{p=0}^n c_{p,n} x^p,$$ $$c_{p,n}= \frac{(-1)^{p + n}(2n - p)!}{2^{n - p}p!(n - p)!}.$$ The coefficient $c_{p,n}$ is OEIS: A001498. The closed form of $b_{p,n}$ still escapes me.