A306778 -id:A306778 - cp's OEIS frontend

A306774 Decimal expansion of Sum_{k>=2} (-1)^k * zeta(k) * zeta(2*k) / k.

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

Offset: 0

Vaclav Kotesovec, Mar 09 2019

			0.63890616261627409120639827814111749882843891313511598351855454505483176209...

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

Cf. A303670, A306769, A306778.

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

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

Equals log(A303670) + A001620 * Pi^2/6.

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.

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)
```

A324597 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^3, n).

Original entry on oeis.org

1, 2, 918, 11592504000, 86712397842439769400000, 3472997049383321958747830928094241894400000, 4152034082374349458781848863476555783741415883758270213129361920000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

In general, for m > 1, Product_{k=1..n} binomial(n + 1/k^m, n) ~ n^Zeta(m) / c(m), where c(m) = Product_{j>=1} Gamma(1 + 1/j^m).

Equivalently, c(m) = -gamma * Zeta(m) + Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(m*k)/k, where gamma is the Euler-Mascheroni constant A001620.

Cf. A306760, A324596, A306778.

a:= n-> n!^(4*n)*mul(binomial(n+1/k^3, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(4*n) * Product[Binomial[n + 1/j^3, n], {j, 1, n}], {n, 1, 8}]

a(n) ~ n!^(4*n) * n^Zeta(3) / (Product_{j>=1} Gamma(1 + 1/j^3)).

a(n) ~ n^(4*n^2 + 2*n + Zeta(3)) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Zeta(3) + c), where c = A306778 = Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(3*k)/k.

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

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A306774 Decimal expansion of Sum_{k>=2} (-1)^k * zeta(k) * zeta(2*k) / k.

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

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

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A324597 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^3, n).

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A324597 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^3, n).