A393361 Decimal expansion of the weight factor for Hermite-Gauss quadrature of degree 5 corresponding to abscissa A393358.

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

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, this sequence

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

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

Links

A.H.M. Smeets, 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 1200*x^2 - 280*x + 3 = 0.

Equals sqrt(Pi)*(7-sqrt(40))/60.

Example

0.01995324205904591320774345859417357486...

Mathematica

Join[{0}, RealDigits[Sqrt[Pi]*(7-Sqrt[40])/60, 10, 88][[1]]] (* Stefano Spezia, Feb 18 2026 *)

Prog

(PARI) sqrt(Pi)*(7-sqrt(40))/60
(PARI) sqrt(Pi)*polrootsreal(Pol([1200, -280, 3]))[1]
(PARI) n = 5; 2^(n-1)*n!*sqrt(Pi)/(n*polhermite(n-1, polrootsreal(polhermite(n))[(n+1)/2+2]))^2

Crossrefs

Cf. A393358.

Sequence in context: A117232 A155995 A347151 * A229191 A387450 A376642

Adjacent sequences: A393358 A393359 A393360 * A393362 A393363 A393364

Keyword

nonn,cons

Author

A.H.M. Smeets, Feb 18 2026

Status

approved