A393366 Decimal expansion of the weight factor for Hermite-Gauss quadrature of degree 6 corresponding to abscissa A393363.

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

Offset

0,2

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, this sequence, A393367

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

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.

Example

0.157067320322856643916311563508378268912314650035...

Mathematica

First[RealDigits[Sqrt[Pi]*Root[10800*#^3 - 5400*#^2 + 405*# - 1 &, 2], 10, 100]] (* Paolo Xausa, Mar 03 2026 *)

Prog

(PARI) sqrt(Pi)*polrootsreal(Pol([10800, -5400, 405, -1]))[2]

Crossrefs

Cf. A393363.

Sequence in context: A388519 A295823 A161018 * A197254 A013706 A195793

Adjacent sequences: A393363 A393364 A393365 * A393367 A393368 A393369

Keyword

nonn,cons

Author

A.H.M. Smeets, Feb 24 2026

Status

approved