%I #33 Mar 02 2023 08:08:51

%S 1,32,55296,7247757312,92771293593600000,141830962344853556428800000,

%T 30619440571316366848044102687129600000,

%U 1077325790213073725701226681195621188514296627200000

%N Discriminant of Hermite polynomials.

%C 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

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

%H Mohammad K. Azarian, <a href="http://www.ijpam.eu/contents/2007-36-2/9/9.pdf">On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials</a>, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hermite_polynomials">Hermite polynomials</a>

%H <a href="/index/He#Hermite">Index entries for sequences related to Hermite polynomials</a>

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

%F 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

%t Table[2^(3n(n-1)/2)Product[k^k,{k,1,n}],{n,1,8}] (* _Indranil Ghosh_, Feb 24 2017 *)

%o (PARI) for(n=1,8, print1(2^(3*n*(n-1)/2)*prod(j=1,n, j^j), ", ")) \\ _G. C. Greubel_, Jun 10 2018

%o (Magma) [Round(2^(3*n*(n-1)/2)*(&*[j^j: j in [1..n]])): n in [1..8]]; // _G. C. Greubel_, Jun 10 2018

%Y Cf. A002109.

%K nonn

%O 1,2

%A _Eric W. Weisstein_