A393367 - OEIS

%I #9 Apr 06 2026 23:39:30

%S 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,

%T 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,

%U 6,9,6,9,0,9,8,1,6,2,8,2,4,1,0,2,4,0,7,0,6,3

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

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

%C    k | zeros                           | corresponding weights for Hermite-Gauss quadrature

%C   ---+---------------------------------+----------------------------------------------------

%C    2 | A010503                         | A019704

%C    3 | 0, A115754                      | 10*A019717, A019708

%C    4 | A393353, A393354                | A393355, A393356

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

%C    6 | A393362, A393363, A393364       | A393365, A393366, this sequence

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, 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.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hermite-GaussQuadrature.html">Hermite-Gauss Quadrature</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

%e 0.0045300099055088456408574725646271509325100719404...

%t 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 *)

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

%Y Cf. A393364, A393904.

%K nonn,cons

%O 0,3

%A _A.H.M. Smeets_, Mar 02 2026