The first Hermite numbers are:
Are obtained from recursion relations of Hermitian polynomials for x = 0:
Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn:
where (n - 1)!! = 1 × 3 × ... × (n - 1).
From the generating function of Hermitian polynomials it follows that
where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. (See Umbral calculus.)
- Weisstein, Eric W. "Hermite Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html