A394167 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.

1, 1, 2, 6, 48, 400, 10800, 246960, 21073920, 1316818944, 342921600000, 58442186880000, 45444644517888000, 21101743065006489600, 48272757747179554068480, 61034174848925197920000000, 406342672548486275137536000000, 1398378373047164966922973347840000

Offset

0,3

Comments

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.

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..94

Eric Weisstein's World of Mathematics, Hermite-Gauss Quadrature.

Formula

a(n) = A393904(n,0)/abs(A393904(n,floor(n/2))).

Crossrefs

Cf. A393904.

Cf. A391956 (Laguerre-Gauss).

Sequence in context: A228159 A249786 A292934 * A195203 A365285 A052743

Adjacent sequences: A394164 A394165 A394166 * A394168 A394169 A394170

Keyword

nonn

Author

A.H.M. Smeets, Mar 11 2026

Status

approved