cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

Examples

			0.63890616261627409120639827814111749882843891313511598351855454505483176209...

Links

Crossrefs

Cf. A303670, A306769, A306778.

Programs

  • Maple
    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)

Formula

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

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

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Apr 28 2018

Keywords

Examples

			0.73302494338583016910945992884780993498453383505001022198223...

Crossrefs

Cf. A306769, A306774, A324590.

Programs

  • Maple
    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
  • Mathematica
    RealDigits[NProduct[Gamma[1 + 1/n^2], {n, 1, Infinity}, WorkingPrecision -> 120, NProductFactors -> 1000], 10, 70][[1]]
  • PARI
    exp(-Euler*Pi^2/6 + sumalt(k=2, (-1)^k*zeta(k)*zeta(2*k)/k)) \\ Vaclav Kotesovec, Mar 09 2019

Formula

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).

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

Views

Author

Vaclav Kotesovec, Mar 09 2019

Keywords

Examples

			0.590735855511984251590434820597746794429756999963932327463401417614129...

Links

Crossrefs

Cf. A306769, A306774, A324597.

Programs

  • Maple
    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)

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

Views

Author

Vaclav Kotesovec, Mar 08 2019

Keywords

Examples

			2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

Links

Crossrefs

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

Programs

  • Maple
    evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);
  • Mathematica
    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}]
  • PARI
    exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

Formula

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).

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

Views

Author

Vaclav Kotesovec, Mar 25 2019

Keywords

Comments

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

Examples

			1.8369219085956637832656408801121703431625646624535449044570417275970793489...

Links

Crossrefs

Cf. A306769, A306774, A306778, A306907.

Programs

  • Maple
    evalf(Sum((-1)^j*Zeta(j)^3/j, j=2..infinity), 120);
  • Mathematica
    NSum[(-1)^k*Zeta[k]^3/k, {k, 2, Infinity}, WorkingPrecision -> 200, NSumTerms -> 100000]
  • PARI
    sumalt(k=2, (-1)^k*zeta(k)^3/k)
Showing 1-5 of 5 results.

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