A393371 Decimal expansion of the weight factor for Hermite-Gauss quadrature of degree 7 corresponding to abscissa 0.

8, 1, 0, 2, 6, 4, 6, 1, 7, 5, 5, 6, 8, 0, 7, 3, 2, 6, 7, 6, 4, 8, 7, 6, 5, 6, 3, 8, 1, 3, 0, 9, 4, 9, 4, 0, 7, 0, 7, 4, 5, 1, 1, 7, 9, 9, 4, 1, 6, 6, 2, 6, 8, 7, 1, 8, 3, 4, 5, 4, 9, 8, 9, 6, 4, 7, 0, 4, 5, 1, 5, 8, 6, 7, 0, 1, 8, 6, 1, 4, 0, 0, 5, 7, 1, 2, 0, 3

Offset

0,1

Comments

There are floor(k/2) positive zeros of the Hermite polynomial of degree k:

k | zeros | corresponding weights for Hermite-Gauss quadrature

---+---------------------------------+----------------------------------------------------

2 | A010503 | A019704

3 | 0, A115754 | 10*A019717, A019708

4 | A393353, A393354 | A393355, A393356

5 | 0, A393357, A393358 | A245887, A393360, A393361

6 | A393362, A393363, A393364 | A393365, A393366, A393367

7 | 0, A393368, A393369, A393370 | this sequence, A393372, A393373, A393374

Links

Table of n, a(n) for n=0..87.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. Eq. 25.4.46 p. 890 and Table 25.10 p. 924.

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

Formula

Equals 16 * sqrt(Pi) / 35.

Equals 2^6 * 7! * sqrt(Pi) / (7*A067994(6))^2.

Equals (A046161(3) / A001803(3)) * sqrt(Pi).

Example

0.81026461755680732676487656381309494070745117994...

Mathematica

RealDigits[16/35*Sqrt[Pi], 10, 100][[1]] (* Amiram Eldar, Mar 07 2026 *)

Prog

(PARI) 16/35*sqrt(Pi)
(PARI) n = 7; 2^(n-1)*n!*sqrt(Pi)/(n*polhermite(n-1, 0))^2

Crossrefs

Cf. A001803, A046161, A067994.

Sequence in context: A230049 A088990 A351129 * A214097 A357468 A373508

Adjacent sequences: A393368 A393369 A393370 * A393372 A393373 A393374

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 07 2026

Status

approved