A391956 a(n) = 1/Product_{i = 1..n} w_i, where w_i is the i-th weight factor for Laguerre-Gauss quadrature of degree n.

1, 1, 8, 486, 221184, 750000000, 18895680000000, 3531604429335150000, 4891710928142129430528000, 50181356668275133180188221767680, 3810801545415429046272000000000000000000, 2141591602439723039788666936461886800000000000000, 8904178923323813039189941618529553910755974184960000000000

Offset

0,3

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..48

A.H.M. Smeets, Table of n, log(a(n)) for some large values of n

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

Formula

Limit_{n -> oo} log(a(n))/n^2 = 1.

a(n) = Product_{k = 1..A000720(n)} A000040(k)^A391957(n,k).

a(n) = (-1)^n*A387347(n,0)/A387347(n,n).

From Vaclav Kotesovec, Jan 16 2026: (Start)

a(n) ~ A^4 * exp(n^2 + n - 1/6) / ((2*Pi)^(n + 1/2) * n^(1/6)), where A is the Glaisher-Kinkelin constant A074962.

a(n) = Product_{k=1..n} k^(4*k - 2*n - 1).

a(n) = (Product_{k=1..n} k^k)^4 / n!^(2*n+1). (End)

Limit_{n -> oo} (log(a(n))-n^2)/n = -log(2*Pi/e) = 1-A061444, discovered by Vaclav Kotesovec. - A.H.M. Smeets, Jan 17 2026

Mathematica

Table[Product[k^(4*k - 2*n - 1), {k, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 16 2026 *)
(* or *)
Round[Table[xi = x /. Solve[LaguerreL[n, x] == 0]; wi = Table[xi[[j]]/((n + 1)^2*LaguerreL[n + 1, xi[[j]]]^2), {j, 1, n}]; 1/Product[wi[[j]], {j, 1, n}], {n, 0, 15}]] (* Vaclav Kotesovec, Jan 16 2026 *)

Crossrefs

Cf. A061444, A387347, A391957.

Sequence in context: A332148 A109060 A259926 * A282180 A240398 A265231

Adjacent sequences: A391953 A391954 A391955 * A391957 A391958 A391959

Keyword

nonn

Author

A.H.M. Smeets, Dec 23 2025

Status

approved