The numbers , where is a Hermite polynomial, may be called Hermite numbers. For , 1, ..., the first few are 1, 0, , 0, 12, 0, , 0, 1680, 0, ... (OEIS A067994). They are given explicitly by
(1)
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(2)
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As a result of the ratio always being divisible by for , the only prime Hermite number is .
The Hermite numbers are related to the Hermite polynomials by
(3)
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where , and
(4)
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where .