A306907 -id:A306907 - cp's OEIS frontend

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

1, 2, 90, 705600, 4105057320000, 52487876090562232320000, 3487017405172854771910634342400000000, 2448893405298238642974553493547144534294528000000000000, 33257039167768610289435138215602132823918399655132218973388800000000000000000

Offset: 0

Vaclav Kotesovec, Mar 08 2019

nonn

Cf. A079478, A101686, A185141, A324444, A306765, A324589, A306907.

a:= n-> mul(mul(i*j+1, i=1..n), j=1..n):
seq(a(n), n=0..9);  # Alois P. Heinz, Jun 24 2023

Table[Product[i*j + 1, {i, 1, n}, {j, 1, n}], {n, 1, 10}]
Table[n!^(2*n) * Product[Binomial[n + 1/j, n], {j, 1, n}], {n, 1, 10}]

a(n) ~ c * n^(n*(2*n+1) + 2*gamma) * (2*Pi)^n * exp(1/6 + log(n)^2 - 2*n^2), where c = 1/A306765 and gamma is the Euler-Mascheroni constant A001620.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A307106 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^3 / k.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 25 2019

Sum_{k>=2} (-1)^k*Zeta(k)/k = A001620 (see MathWorld, formula 122).

			1.8369219085956637832656408801121703431625646624535449044570417275970793489...

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A306769, A306774, A306778, A306907.

evalf(Sum((-1)^j*Zeta(j)^3/j, j=2..infinity), 120);

NSum[(-1)^k*Zeta[k]^3/k, {k, 2, Infinity}, WorkingPrecision -> 200, NSumTerms -> 100000]

PARI
```
sumalt(k=2, (-1)^k*zeta(k)^3/k)
```

A325051 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i!j!k! + 1).

Original entry on oeis.org

2, 256, 19131876000000, 20879156515576282948808247752954619590255260568062500000000

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Next term is too long to be included.

Cf. A306907, A325049.

Table[Product[i!*j!*k! + 1, {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 5}]
Table[BarnesG[n+2]^(3*(n+1)^2) * Product[1 + 1/(i!*j!*k!), {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 5}]

a(n) ~ c * (2*Pi)^(3*n^3/2 + 9*n^2/2 + 9*n/2 + 3/2) * n^((n+1)^2*(6*n^2 + 12*n + 5)/4) / (A^(3*(n+1)^2) * exp(9*n^4/4 + 15*n^3/2 + 8*n^2 + 9*n/4 - 59/80)), where A is the Glaisher-Kinkelin constant A074962 and c = Product_{i>=0, j>=0, k>=0} (1 + 1/(i!*j!*k!)) = 10013049.64089403856780758322163675337812476527762657951330...

cp's OEIS Frontend

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

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A307106 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^3 / k.

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A325051 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i!j!k! + 1).

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cp's OEIS Frontend

A306760 a(n) = Product_{i=1..n, j=1..n} (i*j + 1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A307106 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^3 / k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A325051 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i!*j!*k! + 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A325051 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i!j!k! + 1).