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I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k,n}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=Touchard \ polynomials$.

I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k,n}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=$ Touchard polynomials.

I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=Touchard \ polynomials$.

I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k,n}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=Touchard \ polynomials$.

I know this will sound like a general question, but given the polynomial sequence $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=Touchard \ polynomials$.

I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what are some some of its properties - including recurrence relations and generating functions? Is it of bynomial type? I know that when $a_{k}={k}^{n}$, we have $P_{n}(x)=T_{n}(x)=Touchard \ polynomials$.