A382686 Decimal expansion of the weight factor for Legendre-Gauss quadrature corresponding to abscissa A372272.

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

Offset

0,1

Comments

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:

k | zeros | corresponding weights

---+---------------------------+--------------------------

2 | A020760 | A000007*10

3 | A010513/10 | A010716

4 | A372267, A372268 | A382103, A382104

5 | A372269, A372270 | A382105, A382106

6 | A372271, A372272, A372273 | A382107, this sequence, A382687

7 | A372274, A372275, A372276 | A382688, A382689, A382690

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]. Table 25.4, n=6.

A.H.M. Smeets, Python program for Legendre-Gauss quadrature constants.

Eric Weisstein's World of Mathematics, Legendre-Gauss Quadrature.

Example

0.36076157304813860756983351383771611166152189274674...

Mathematica

RealDigits[With[{x = Root[LegendreP[6, #] &, 5]}, 2*(1-x^2)/(49*LegendreP[7, x]^2)], 10, 120][[1]] (* Amiram Eldar, Apr 14 2026 *)

Crossrefs

Cf. A372272.

Sequence in context: A181916 A077590 A199182 * A011368 A020811 A200005

Adjacent sequences: A382683 A382684 A382685 * A382687 A382688 A382689

Keyword

nonn,cons

Author

A.H.M. Smeets, Apr 03 2025

Status

approved