The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #13 Mar 29 2026 14:48:54
%S 1,1,2,6,48,400,10800,246960,21073920,1316818944,342921600000,
%T 58442186880000,45444644517888000,21101743065006489600,
%U 48272757747179554068480,61034174848925197920000000,406342672548486275137536000000,1398378373047164966922973347840000
%N a(n) = 1/Product_{i = 1..floor(n/2)} W_i, where W_i is the i-th normalized weight factor for Hermite-Gauss quadrature of degree n, corresponding to the i-th positive abscissa.
%C In Hermite-Gauss quadrature, the weighting function is given by w(x) = exp(-x^2) over the interval (-oo,oo), and thus Integral_{x=-oo..oo} w(x) dx = sqrt(Pi). The normalized weighting function W(x) and weighting factors W_i are given such that Integral_{x=-oo..oo} W(x) dx = 1; i.e., W(x) = w(x)/sqrt(Pi) and W_i = w_i/sqrt(Pi). The normalized weighting factors W_i of degree n are algebraic numbers of degree floor(n/2) if the corresponding abscissa is not zero.
%H A.H.M. Smeets, <a href="/A394167/b394167.txt">Table of n, a(n) for n = 0..94</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hermite-GaussQuadrature.html">Hermite-Gauss Quadrature</a>.
%F a(n) = A393904(n,0)/abs(A393904(n,floor(n/2))).
%Y Cf. A393904.
%Y Cf. A391956 (Laguerre-Gauss).
%K nonn
%O 0,3
%A _A.H.M. Smeets_, Mar 11 2026