A391956 - OEIS

%I #43 Jul 11 2026 18:43:36

%S 1,1,8,486,221184,750000000,18895680000000,3531604429335150000,

%T 4891710928142129430528000,50181356668275133180188221767680,

%U 3810801545415429046272000000000000000000,2141591602439723039788666936461886800000000000000,8904178923323813039189941618529553910755974184960000000000

%N 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.

%H A.H.M. Smeets, <a href="/A391956/b391956.txt">Table of n, a(n) for n = 0..48</a>

%H A.H.M. Smeets, <a href="/A391956/a391956.txt">Table of n, log(a(n)) for some large values of n</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Laguerre-GaussQuadrature.html">Laguerre-Gauss Quadrature</a>.

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

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

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

%F From _Vaclav Kotesovec_, Jan 16 2026: (Start)

%F 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.

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

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

%F 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

%t Table[Product[k^(4*k - 2*n - 1), {k, 1, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jan 16 2026 *)

%t (* Alternative: *)

%t 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 *)

%Y Cf. A061444, A387347, A391957.

%K nonn,changed

%O 0,3

%A _A.H.M. Smeets_, Dec 23 2025