A100072 Decimal expansion of Product_{n>=1} (1/e * (1/(3*n)+1)^(3*n+1/2)).

1, 0, 1, 2, 3, 7, 8, 5, 5, 2, 7, 2, 2, 9, 1, 2, 2, 4, 9, 5, 3, 9, 6, 0, 2, 9, 6, 0, 4, 9, 6, 6, 8, 8, 6, 9, 2, 9, 7, 8, 0, 4, 4, 8, 7, 5, 8, 6, 9, 1, 7, 7, 1, 5, 0, 2, 8, 2, 0, 2, 2, 6, 5, 9, 5, 9, 2, 9, 3, 5, 4, 3, 2, 4, 3, 1, 0, 7, 8, 0, 9, 2, 3, 4, 6, 6, 1, 5, 9, 2, 9, 7, 4, 0, 3, 1, 1, 5, 8, 6, 8, 2

Offset

1,4

Links

Table of n, a(n) for n=1..102.

Mathematics Stackexchange, Proof of a closed-form of Product_{n=1..oo} 1/e * (1/(3*n)+1)^(3*n+1/2), Dec 28, 2021.

Eric Weisstein's World of Mathematics, Infinite Product

Formula

3^(13/24) * exp(1 + (2*Pi^2 - 3*PolyGamma(1, 1/3))/(12*sqrt(3)*Pi)) * sqrt(Gamma(1/3)/(2*Pi)) / A^4, where A = A074962 is the Glaisher-Kinkelin constant.

Example

1.012378552722912249539602960496688692978044875869177150282...

Maple

evalf(product(exp(-1)*(1/(3*n)+1)^(3*n+1/2), n = 1..infinity), 104); # Vaclav Kotesovec, Aug 16 2015

Mathematica

RealDigits[(3^(13/24)*E^(1 + (2*Pi^2 - 3*PolyGamma[1, 1/3])/(12*Sqrt[3]*Pi)) * Sqrt[Gamma[1/3]/(2*Pi)])/Glaisher^4, 10, 100][[1]] (* Vaclav Kotesovec, Aug 16 2015  after Eric W. Weisstein *)
N[Product[1/E*(1/(3*n) + 1)^(3*n + 1/2), {n, 1, Infinity}], 101] (* Vaclav Kotesovec, Aug 16 2015 *)

Crossrefs

Cf. A074962.

Sequence in context: A199466 A199966 A011027 * A215722 A324777 A244162

Adjacent sequences: A100069 A100070 A100071 * A100073 A100074 A100075

Keyword

nonn,cons

Author

Eric W. Weisstein, Nov 02 2004

Status

approved