A263416 - OEIS

%I #29 Feb 16 2025 08:33:27

%S 1,1,4,112,31360,114150400,6648119296000,7356542888181760000,

%T 179090034163620983603200000,108995627512253039588776345600000000,

%U 1857397104331364341705287836001894400000000000,981210407605679794692064339146706741991833600000000000000

%N a(n) = Product_{k=0..n} (3*k+1)^(n-k).

%H G. C. Greubel, <a href="/A263416/b263416.txt">Table of n, a(n) for n = 0..50</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygammaFunction.html">Polygamma Function</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.

%F a(n) ~ A^(1/3) * 2^(n/2 + 1/6) * 3^(n^2/2 + n/2 - 1/72) * n^(n^2/2 + n/3 - 1/36) * Pi^(n/2 + 1/6) / (Gamma(1/3)^(n + 1/3) * exp(3*n^2/4 + n/3 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) + 1/36)), where A = A074962 is the Glaisher-Kinkelin constant and PolyGamma(1, 1/3) = 10.09559712542709408179200409989... (PolyGamma[1, 1/3] in Mathematica or Psi(1, 1/3) in Maple).

%F PolyGamma(1, 1/3) = 3^(3/2) * A261024 + 2*Pi^2/3.

%F From _Vladimir Reshetnikov_, Nov 11 2015: (Start)

%F a(n) = 3^(n*(n+1)/2) * G(n+4/3) / (G(1/3) * Gamma(1/3)^(n+1)), where G(x) is the Barnes G-function.

%F a(n) ~ 3^(n*(n+1)/2) * exp(-(9*n^2+4*n-1)/12) * n^((18*n^2+12*n-1)/36) * (2*Pi)^((3*n+1)/6) / (A * G(1/3) * Gamma(1/3)^(n+1)).

%F Note that G(1/3) = 3^(1/72) * exp(1/9 + Pi/(18*sqrt(3)) - PolyGamma(1, 1/3)/(12*sqrt(3)*Pi)) / (A^(4/3) * Gamma(1/3)^(2/3)).

%F (End)

%p A263416:=n->mul((3*k+1)^(n-k), k=0..n): seq(A263416(n), n=0..11); # _Wesley Ivan Hurt_, Nov 12 2015

%t Table[Product[(3*k+1)^(n-k),{k,0,n}],{n,0,12}] (* or *)

%t Table[1/FullSimplify[(Gamma[1/3]^((v-1)/3) / 3^((v-1)/18)) * Exp[Integrate[(E^((3-v)*x) - E^(2*x))/(x*(E^(3*x)-1)^2) + (v-1) * (1/(3*x*(E^(3*x)-1)) + 1/(6*x*E^(3*x)) - (v+1)/(18*x*E^x)), {x, 0, Infinity}]]], {v, 1, 34, 3}]

%t Round@Table[3^(n(n+1)/2) BarnesG[n+4/3]/(BarnesG[1/3] Gamma[1/3]^(n+1)), {n, 0, 12}] (* _Vladimir Reshetnikov_, Nov 11 2015 *)

%o (PARI) a(n) = prod(k=0, n, (3*k+1)^(n-k)); \\ _Michel Marcus_, Nov 12 2015

%Y Cf. A057863, A074962, A252798, A261024, A263405, A263414, A263417, A369468.

%K nonn,easy,changed

%O 0,3

%A _Vaclav Kotesovec_, Oct 17 2015