A393360 - OEIS

A393360

Decimal expansion of the weight factor for Hermite-Gauss quadrature of degree 5 corresponding to abscissa A393357.

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

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OFFSET

0,1

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, this sequence, A393361

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 . 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

0.3936193231522411598284956208520936344548582290546379...

MATHEMATICA

First[RealDigits[Sqrt[Pi]*(7 + Sqrt[40])/60, 10, 100]] (* Paolo Xausa, Feb 23 2026 *)

PROG

(PARI) sqrt(Pi)*(7+sqrt(40))/60

(PARI) sqrt(Pi)*polrootsreal(Pol([1200, -280, 3]))[2]