A393904 Irregular table T(n,m): coefficients of the minimal polynomial for the normalized weight factors of the Hermite-Gauss quadrature (exponents in decreasing order).

1, 1, 2, -1, 6, -1, 48, -24, 1, 1200, -280, 3, 10800, -5400, 405, -1, 1234800, -335160, 9303, -5, 948326400, -474163200, 46005120, -404768, 45, 230443315200, -68401428480, 2998332288, -7898400, 175, 9601804800000, -4800902400000, 540101520000, -8977500000, 6534225, -28

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

Table of n, a(n) for n=0..35.

Example

The first minimal polynomial for the normalized weight factors of the Hermite-Gauss quadrature, corresponding to the positive real abscissa are 1 = 0 (n = 0), 1 = 0 (n = 1), 2*x - 1 = 0 (n = 2), 6*x - 1 = 0 (n = 3), 48*x^2 - 24*x + 1 (n = 4), so the sequence starts with 1, 1, 1, -2, 1, -6, 48, -24, 1. Note that 1 = 0 refers to an empty set of roots.
Triangle starts:
  n\m        0           1         2        3   4
  0          1
  1          1
  2          2          -1
  3          6          -1
  4         48         -24         1
  5       1200        -280         3
  6      10800       -5400       405       -1
  7    1234800     -335160      9303       -5
  8  948326400  -474163200  46005120  -404768  45

Crossrefs

Cf. A112734 (Legendre-Gauss), A387347 (Laguerre-Gauss).

Sequence in context: A369925 A055878 A331654 * A346864 A302690 A030304

Adjacent sequences: A393901 A393902 A393903 * A393905 A393906 A393907

Keyword

sign,tabf

Author

A.H.M. Smeets, Mar 02 2026

Status

approved