Hermite numbers

The Hermite numbers $H_n$  may be defined by the generating function

$e^{-t^2}:={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{H}_n{\displaystyle \frac{t^n}{n!}}$(1)

which is same as the generating function of Hermite polynomials at the value 0 of the argument $z$.  After expanding the left hand side of (1) to Taylor series $1-\frac{t^2}{1!}+\frac{t^4}{2!}-\frac{t^6}{3!}+-\mathrm{\dots }$, one can write

$1-2\cdot {\displaystyle \frac{t^2}{2!}}+12\cdot {\displaystyle \frac{t^4}{4!}}-120\cdot {\displaystyle \frac{t^6}{6!}}+-\mathrm{\dots }={\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{H}_n{\displaystyle \frac{t^n}{n!}}.$(2)

Thus one sees that

$$H_0=1,H_1=0,H_2=-2,H_3=0,H_4=12,H_5=0,H_6=-120,\mathrm{\dots }$$

Evidently,

$H_n=\{\begin{array}{cc}\frac{{(-1)}^{\frac{n}{2}}n!}{(\frac{n}{2})!}\text{ when }2\mid n\hfill & \\ 0\mathit{\hspace{1em}\hspace{1em}\hspace{0.25em}}\text{ when }2\nmid n\hfill & \end{array}$(3)

The Hermite numbers form the sequence (http://www.research.att.com/ njas/sequences/index.html?q=A067994&language=english&go=SearchSloane A067994)

$$1,\mathrm{\hspace{0.17em}0},-2,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}12},\mathrm{\hspace{0.17em}0},-120,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}1680},\mathrm{\hspace{0.17em}0},-30240,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}665280},\mathrm{\hspace{0.17em}0},-17297280,\mathrm{\hspace{0.17em}0},\mathrm{\dots }$$

which obeys the recurrence relation

$H_n=\mathrm{\hspace{0.33em}2}(1-n)H_{n-2}.$(4)

According to (1), the Hermite numbers satisfy  $H_n=H_n(0)$  where $H_n(x)$ is the Hermite polynomial of degree $n$.  The of Hermite numbers and Hermite polynomials may be expressed also by using symbolic powers

$$H^{\nu }=:H_{\nu }$$

as follows:

${(2x+H)}^n=H_n(x).$(5)

This means e.g. that

$${(2x+H)}^2={(2x)}^2+2\cdot 2x{H}^1+H^2=\mathrm{\hspace{0.33em}4}{x}^2+4x{H}_1+H_2=\mathrm{\hspace{0.33em}4}{x}^2-2=H_2(x).$$