A393367 Decimal expansion of the weight factor for Hermite-Gauss quadrature of degree 6 corresponding to abscissa A393364.

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

Offset

0,3

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, this sequence

7 | 0, A393368, A393369, A393370 | A393371, A393372, A393373, A393374

Links

Table of n, a(n) for n=0..89.

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.

Index entries for transcendental numbers.

Formula

This constant divided by sqrt(Pi) is a root of 10800*x^3 - 5400*x^2 + 405*x - 1 = 0 (see also A393904).

Example

0.0045300099055088456408574725646271509325100719404...

Mathematica

With[{nd = 100}, RealDigits[Sqrt[Pi] * x /. FindRoot[10800*x^3 - 5400*x^2 + 405*x - 1, {x, 0}, WorkingPrecision -> nd], 10, nd, -1][[1]]] (* Amiram Eldar, Mar 02 2026 *)

Prog

(PARI) sqrt(Pi)*polrootsreal(Pol([10800, -5400, 405, -1]))[1]

Crossrefs

Cf. A393364, A393904.

Sequence in context: A354700 A046572 A046574 * A019743 A010663 A385445

Adjacent sequences: A393364 A393365 A393366 * A393368 A393369 A393370

Keyword

nonn,cons

Author

A.H.M. Smeets, Mar 02 2026

Status

approved