A054374 Discriminant of Hermite polynomials.

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

Offset

1,2

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

Table of n, a(n) for n=1..8.

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

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

Mathematica

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

Prog

(PARI) for(n=1, 8, print1(2^(3*n*(n-1)/2)*prod(j=1, n, j^j), ", ")) \\ G. C. Greubel, Jun 10 2018
(Magma) [Round(2^(3*n*(n-1)/2)*(&*[j^j: j in [1..n]])): n in [1..8]]; // G. C. Greubel, Jun 10 2018

Crossrefs

Cf. A002109.

Sequence in context: A159396 A362175 A221086 * A221138 A017009 A176030

Adjacent sequences: A054371 A054372 A054373 * A054375 A054376 A054377

Keyword

nonn

Author

Eric W. Weisstein

Status

approved