A306774 -id:A306774 - cp's OEIS frontend

A303670 Decimal expansion of Product_{k>=1} Gamma(1 + 1/k^2).

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

Offset: 0

Vaclav Kotesovec, Apr 28 2018

			0.73302494338583016910945992884780993498453383505001022198223...

Cf. A306769, A306774, A324590.

Digits := 120: evalf(product(GAMMA(1+1/n^2), n = 1..infinity));
evalf(exp(-gamma*Pi^2/6 + Sum((-1)^k*Zeta(k)*Zeta(2*k)/k, k=2..infinity)), 121); # Vaclav Kotesovec, Mar 09 2019

RealDigits[NProduct[Gamma[1 + 1/n^2], {n, 1, Infinity}, WorkingPrecision -> 120, NProductFactors -> 1000], 10, 70][[1]]

exp(-Euler*Pi^2/6 + sumalt(k=2, (-1)^k*zeta(k)*zeta(2*k)/k)) \\ Vaclav Kotesovec, Mar 09 2019

Equals Product_{k>=1} Gamma(1/k^2) / k^2.
Equals exp(-gamma*Pi^2/6 + Sum_{k>=2} (-1)^k*zeta(k)*zeta(2*k)/k), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 09 2019
Equals exp(-gamma*Pi^2/6 + A306774).

A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 09 2019

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

			1.043402917574288733255289646671676030548470866046882561044570479769585...

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

Cf. A231132, A306760, A306765, A306774, A306778, A307106.

evalf(Sum((-1)^j*Zeta(j)^2/j, j=2..infinity), 100);

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

sumalt(k=2, (-1)^k*zeta(k)^2/k) \\ Michel Marcus, Mar 09 2019

Equals log(A306765) + A001620^2.

A306778 Decimal expansion of Sum_{k>=2} (-1)^k * zeta(k) * zeta(3*k) / k.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Mar 09 2019

			0.590735855511984251590434820597746794429756999963932327463401417614129...

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

Cf. A306769, A306774, A324597.

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

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

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)
```

A324590 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 1080, 16133644800, 139256878046022696960000, 6288402750181849898716908922601472000000000, 8322157105451357856813375261666887975745751468393837363200000000000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

Cf. A303670, A306760, A306774, A324589.

a:= n-> n!^(4*n)*mul(binomial(n+1/k^2, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(4*n) * Product[Binomial[1/k^2 + n, n], {k, 1, n}], {n, 1, 8}]

a(n) ~ n!^(4*n) * n^(Pi^2/6) / A303670.
a(n) ~ n^(4*n^2 + 2*n + Pi^2/6) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.
a(n) = n!^n * A324596(n).

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

A324596 a(n) = n!^(3n) Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 270, 74692800, 419731620267960000, 252716802910471719823692648960000, 59736659298524125157504488525739821430187940800000000, 16079377413231597423103950774423398920424350187193326745026311068057600000000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

Cf. A303670, A306760, A306774, A324589, A324597.

a:= n-> n!^(3*n)*mul(binomial(n+1/k^2, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(3*n) * Product[Binomial[n + 1/k^2, n], {k, 1, n}], {n, 1, 8}]

a(n) ~ n!^(3*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(3*n*(2*n+1)/2 + Pi^2/6) * (2*Pi)^(3*n/2) / exp(3*n^2 - 1/4 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

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

cp's OEIS Frontend

A303670 Decimal expansion of Product_{k>=1} Gamma(1 + 1/k^2).

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A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

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A306778 Decimal expansion of Sum_{k>=2} (-1)^k * zeta(k) * zeta(3*k) / k.

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

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A324590 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

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A324596 a(n) = n!^(3*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

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A324590 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^2, n).

A324596 a(n) = n!^(3n) Product_{k=1..n} binomial(n + 1/k^2, n).