A054374 Discriminant of Hermite polynomials.

1, 32, 55296, 7247757312, 92771293593600000, 141830962344853556428800000, 30619440571316366848044102687129600000, 1077325790213073725701226681195621188514296627200000

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

Comments

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego eq. (6.71.7). - Alan Sokal, Mar 02 2012

References

• G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

Links

• Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
• Eric Weisstein's World of Mathematics, Hermite Polynomial.
• Wikipedia, Hermite polynomials
• Index entries for sequences related to Hermite polynomials

Crossrefs

Cf. A002109.

Programs

Magma

[Round(2^(3*n*(n-1)/2)*(&*[j^j: j in [1..n]])): n in [1..8]]; // G. C. Greubel, Jun 10 2018

Mathematica

Table[2^(3n(n-1)/2)Product[k^k,{k,1,n}],{n,1,8}] (* Indranil Ghosh, Feb 24 2017 *)

PARI

for(n=1,8, print1(2^(3*n*(n-1)/2)*prod(j=1,n, j^j), ", ")) \\ G. C. Greubel, Jun 10 2018

Formula

a(n) = 2^(3*n*(n-1)/2) * Product_{k=1..n} k^k.

a(n) ~ A * 2^(3*n*(n-1)/2) * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Mar 02 2023