A263416 a(n) = Product_{k=0..n} (3*k+1)^(n-k).

1, 1, 4, 112, 31360, 114150400, 6648119296000, 7356542888181760000, 179090034163620983603200000, 108995627512253039588776345600000000, 1857397104331364341705287836001894400000000000, 981210407605679794692064339146706741991833600000000000000

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..50

Eric Weisstein's World of Mathematics, Glaisher-Kinkelin Constant

Eric Weisstein's World of Mathematics, Polygamma Function

Eric Weisstein's World of Mathematics, Barnes G-Function.

Wikipedia, Barnes G-function.

Formula

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).

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

From Vladimir Reshetnikov, Nov 11 2015: (Start)

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.

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)).

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)).

(End)

Maple

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

Mathematica

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

Prog

(PARI) a(n) = prod(k=0, n, (3*k+1)^(n-k)); \\ Michel Marcus, Nov 12 2015

Crossrefs

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

Sequence in context: A196458 A262261 A085522 * A078780 A041377 A103499

Adjacent sequences: A263413 A263414 A263415 * A263417 A263418 A263419

Keyword

nonn,easy

Author

Vaclav Kotesovec, Oct 17 2015

Status

approved