A367842 -id:A367842 - cp's OEIS frontend

A367958 a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

1, 6, 5544, 2822916096, 1723467782592331776, 2210440498434925488635904000000, 9234659938893939743399592700454853672960000000, 180150216814109052335771891722360520401032374209013927116800000000

Offset: 0

Vaclav Kotesovec, Dec 06 2023

nonn

In general, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^(1/d) * (Product_{j=1..d} Gamma(j/d)^(j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*n*(n+1) + (d+1)^2/(12*d) + 1/12) * n^(n^2 - d/12 - 1/4 - 1/(12*d)) / ((2*Pi)^((d+1)/4) * exp(3*n^2/2 + 1/(12*d)) * d^((n*(d*n + (d+1)))/2 - 1/(12*d))), where A = A074962 is the Glaisher-Kinkelin constant.

Equivalently, for d>0, Product_{i=1..n, j=1..n} (i + d*j) ~ A^d * (Product_{j=1..d} BarnesG(j/d)) * (2*Pi)^((d-3)/4) * (d+1)^((d + (d+1)^2*(6*n*(n+1) + 1)) / (12*d)) * n^(n^2 - 1/4 - 1/(12*d) - d/12) / (d^((n+1)*(d*n + 1)/2) * exp(3*n^2/2 + d/12)).

Cf. A079478 (d=1), A324402 (d=2), A367956 (d=3), A367957 (d=4).

Cf. A074962, A367842, A367944.

a:= n-> mul(mul(i+5*j, i=1..n), j=1..n):
seq(a(n), n=0..8);  # Alois P. Heinz, Dec 06 2023

Table[Product[i + 5*j, {i, 1, n}, {j, 1, n}], {n, 0, 10}]

a(n) ~ A^(1/5) * (1 + sqrt(5))^(1/10) * 2^(18*n*(n+1)/5 + 29/60) * 3^(18*n*(n+1)/5 + 41/60) * n^(n^2 - 41/60) / (Pi^(1/10) * Gamma(1/5)^(3/5) * Gamma(2/5)^(1/5) * 5^(n*(5*n+6)/2 + 1/3) * exp(3*n^2/2 + 1/60)), where A = A074962 is the Glaisher-Kinkelin constant.

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).

A367899 Decimal expansion of limit_{n->oo} Product_{k=1..n} BarnesG(k/n)^(k/n^2).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 04 2023

			0.82679846439497137183536464944643006378339978236702912024106018188...

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

Cf. A074962, A367842, A367898.

RealDigits[E^(1/24 + 3*Zeta[3]/(8*Pi^2))/(Sqrt[Glaisher]*(2*Pi)^(1/12)), 10, 120][[1]]
Exp[Integrate[x*Log[BarnesG[x]], {x, 0, 1}]]

Equals exp(1/24 + 3*zeta(3)/(8*Pi^2)) / (sqrt(A) * (2*Pi)^(1/12)), where A = A074962 is the Glaisher-Kinkelin constant.

Equals exp(Integral_{x=0..1} x*log(BarnesG(x)) dx).

A375368 Decimal expansion of zeta'(2)/(2Pi^2) + log(2Pi)/6 - gamma/12.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 13 2024

zeta'(2) = -0.9375.. is the first derivative of the zeta function (see A073002). Gamma is A001620.

			0.21071478956855210834291187462669484383332902315035...

Olivier Espinosa and Victor H. Moll, On some integrals involving the Hurwitz zeta function: Part 1, Raman. J. 6 (2002) 159-188, Example 6.4.

Cf. A001620, A073002, A367842, A375369.

Zeta(1,2)/2/Pi^2+log(2*Pi)/6-gamma/12 ; evalf(%) ;

RealDigits[Zeta'[2] / (2*Pi^2) + Log[2*Pi] / 6 - EulerGamma / 12, 10, 120][[1]] (* Amiram Eldar, Aug 19 2024 *)

Equals Integral_{x=0..1} x* log(Gamma(x)) dx.

Equals log(A367842). - Hugo Pfoertner, Aug 19 2024

cp's OEIS Frontend

A367958 a(n) = Product_{i=1..n, j=1..n} (i + 5*j).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A367899 Decimal expansion of limit_{n->oo} Product_{k=1..n} BarnesG(k/n)^(k/n^2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A375368 Decimal expansion of zeta'(2)/(2*Pi^2) + log(2*Pi)/6 - gamma/12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A375368 Decimal expansion of zeta'(2)/(2Pi^2) + log(2Pi)/6 - gamma/12.