A367899 -id:A367899 - cp's OEIS frontend

A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

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

Offset: 1

Vaclav Kotesovec, Dec 02 2023

Limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(1/n) = sqrt(2*Pi).

			1.23456019539799897381741853007827189474437277093956302475669920823457...

Cf. A010767, A092040, A074962, A367898, A367899.

RealDigits[(2*Pi)^(1/4)/Glaisher, 10, 120][[1]]
Exp[Integrate[x*Log[Gamma[x]], {x, 0, 1}]]

Equals (2*Pi)^(1/4) / A, where A = A074962 is the Glaisher-Kinkelin constant.
Equals A010767 * A092040 / A074962.
Equals exp(Integral_{x=0..1} x*log(Gamma(x)) dx).

A367898 Decimal expansion of limit_{n->oo} Product_{k=1..n} BarnesG(k/n)^(1/n).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 04 2023

			0.4174272976014098636439484516225169770945963322141100823211317682...

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

Cf. A074962, A367842, A367899.

RealDigits[Exp[1/12] / (Glaisher^2 * (2*Pi)^(1/4)), 10, 120][[1]]

Equals exp(1/12) / (A^2 * (2*Pi)^(1/4)), where A = A074962 is the Glaisher-Kinkelin constant.

Product_{k=1..n} BarnesG(k/n) = A^(1/n - n) * exp((n - 1/n)/12) * n^(1/2 + 1/(12*n)) * (2*Pi)^((1-n)/2) * Product_{k=1..n-1} Gamma(k/n)^(k/n).

cp's OEIS Frontend

A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

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