A303670 -id:A303670 - 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.

A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 08 2019

			2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

Vaclav Kotesovec, Table of n, a(n) for n = 1..297

Cf. A001620, A231132, A303670, A306760, A306769.

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

slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]

exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.
Equals exp(-gamma^2 + A306769).
Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(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

A363684 Decimal expansion of Product_{k>=1} Gamma(2k/(2k-1)) / Gamma(1+1/(2k)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 15 2023

			1.06215090557105728069683736293...

Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Exp. Math. 9 (1) (2000) 3-12.
Vaclav Kotesovec, Why do I get different results for the products of two identical expressions ?, Mathematica Stack Exchange, Jun 10 2024.

Cf. A303670.

evalf(Product(2*k*GAMMA(1/(2*k - 1))/((2*k - 1)*GAMMA(1/(2*k))), k = 1..infinity), 120); # Vaclav Kotesovec, Jun 10 2024

default(realprecision, 200); exp(sumpos(k=1, log(2*k) + log(gamma(1/(2*k-1))) - log(2*k-1) - log(gamma(1/(2*k))) )) \\ Vaclav Kotesovec, Jun 10 2024

More terms from Vaclav Kotesovec, Jun 10 2024

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

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A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

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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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Maple

Mathematica

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A363684 Decimal expansion of Product_{k>=1} Gamma(2k/(2k-1)) / Gamma(1+1/(2k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

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).