A387347 Coefficients of the minimal polynomial for the weight factors of Laguerre-Gauss quadrature (exponents in decreasing order).

1, 1, -1, 8, -8, 1, 1944, -1944, 405, -4, 1990656, -1990656, 504576, -16960, 9, 3888000000000, -3888000000000, 1094580000000, -65345000000, 223349625, -5184, 944784000000000, -944784000000000, 284616180000000, -23430886200000, 219303946125, -55842372, 50

Offset

0,4

Comments

The weight factors for Laguerre-Gauss quadrature are the n real roots of the minimal polynomial given here. The largest root corresponds to the smallest abscissas, and the smallest root corresponds to the largest abscissas for Laguerre-Gauss quadrature. See also A384463, A384464, A384465 for n = 3, and A384466, A384467, A384588, A384589 for n = 4.

T(n,0) has only prime factors p_i <= n.

T(n,n) has only prime factors p_i < n (see A392806).

Can anyone formulate a recurrence relation for the coefficients of the normalized minimal polynomials (see link "Normalized coefficients"). - A.H.M. Smeets, Feb 12 2026

Links

Table of n, a(n) for n=0..27.

A.H.M. Smeets, Normalized coefficients: (-1)^(n-m)*A387347(n,m)/A387347(n,n) for n = 0..53 and m = 0..n.

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

Formula

T(n,1) = -T(n,0). Proof: Sum_{i = 0..n} w_i = 1, where w_i is the i-th weight factor for Laguerre-Gauss quadrature of degree n.

T(n,0)/T(n,n) = (-1)^n / Product_{i = 1..n} w_i = (-1)^n*A391956(n), w_i as above.

T(n,n) = (-1)^n*A392805(n).

Example

Triangle begins (0 <= k <= n):
  n\k              0                1               2               3            4         5  6
  0:               1
  1:               1               -1
  2:               8               -8               1
  3:            1944            -1944             405              -4
  4:         1990656         -1990656          504576          -16960            9
  5:   3888000000000   -3888000000000   1094580000000    -65345000000    223349625     -5184
  6: 944784000000000 -944784000000000 284616180000000 -23430886200000 219303946125 -55842372 50

Prog

(PARI) default(realprecision, 10000);
L(n, z) = subst(pollaguerre(n), 'x, z);
wl(n) = my(r=polrootsreal(pollaguerre(n))); vector(n, i, r[i]/((n+1)^2*L(n+1, r[i])^2));
np(n) = my(vw=wl(n)); prod(i=1, n, 'x-vw[i])/prod(i=1, n, vw[i]);
row(n) = my(v=apply(bestappr, Vec(np(n)))); v*lcm(apply(denominator, v)); \\ Michel Marcus, Feb 04 2026

Crossrefs

Cf. A384463, A384464, A384465, A384466, A384467, A384588, A384589, A391956, A392805, A392806.

Sequence in context: A351210 A199597 A366149 * A197848 A224875 A242588

Adjacent sequences: A387344 A387345 A387346 * A387348 A387349 A387350

Keyword

sign,tabl

Author

A.H.M. Smeets, Dec 21 2025

Status

approved