A393366 - OEIS

%I #12 Apr 06 2026 23:18:10

%S 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,

%T 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,

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

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

%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, this sequence, A393367

%H Paolo Xausa, <a href="/A393366/b393366.txt">Table of n, a(n) for n = 0..10000</a>

%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.

%e 0.157067320322856643916311563508378268912314650035...

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

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

%Y Cf. A393363.

%K nonn,cons

%O 0,2

%A _A.H.M. Smeets_, Feb 24 2026