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.

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A271854 Decimal expansion of -zeta'(-1/2), negated derivative of the Riemann zeta function at -1/2.

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, Apr 23 2016

Keywords

Examples

			zeta'(-1/2) = -0.36085433959994760734742080636395106588485278791863221...

Links

Crossrefs

Values of |zeta'(x)| for various x: A073002 (+2), A075700 (0), A084448 (-1), A114875 (+1/2), A240966 (-2), A244115(+3), A259068 (-3), A259069 (-4), A259070 (-5), A259071 (-6), A259072 (-7), A259073 (-8), A261506 (+4), A266260 (-9), A266261 (-10), A266262 (zeta'(-11)), A266263 (zeta'(-12)), A260660 (zeta'(-13)), A266264 (zeta'(-14)), A266270 (zeta'(-15)), A266271 (zeta'(-16)), A266272 (zeta'(-17)), A266273 (zeta'(-18)), A266274 (zeta'(-19)), A266275 (zeta'(-20)), A271521 (i).

Programs

  • Mathematica
    RealDigits[N[-Zeta'[-1/2], 106]] [[1]] (* Robert Price, Apr 28 2016 *)
  • PARI
    -zeta'(-1/2)

A372140 a(n) = Product_{k=1..n} BarnesG(k)^k.

Original entry on oeis.org

1, 1, 1, 1, 16, 3981312, 2271857773302207479808, 133781874275586180035265927852035878702421114880000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2024

Keywords

Comments

The next term has 113 digits.

Links

Crossrefs

Cf. A055462, A255269.

Programs

  • Mathematica
    Table[Product[BarnesG[k]^k, {k, 1, n}], {n, 0, 8}]

Formula

a(n) ~ (2*Pi)^(n*(n^2 - 1)/6) * n^(n^4/8 - n^3/12 - n^2/6 + n/24 + 19/720) / (A^(n^2/2 + n/2 - 1/3) * exp(7*n^4/32 - 5*n^3/72 - 7*n^2/24 - n/24 - zeta(3)/(8*Pi^2) + zeta'(-3)/6 + 23/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.

A371339 a(n) = Product_{k=1..n} A000178(k)^k.

Original entry on oeis.org

1, 1, 4, 6912, 47552535724032, 2344457420244640062508151026483200000, 556518660278190472985800630083758030134707790620313895060688076800000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2024

Keywords

Links

Crossrefs

Cf. A000178, A055462, A255269, A372140.

Programs

  • Mathematica
    Table[Product[BarnesG[k+2]^k, {k, 1, n}], {n, 0, 8}]

Formula

a(n) = Product_{k=1..n} BarnesG(k+2)^k.
a(n) = A372140(n+2) / A055462(n)^2.
a(n) ~ (2*Pi)^(n*(n+1)*(n+2)/6) * n^(n^4/8 + 7*n^3/12 + 5*n^2/6 + 3*n/8 + 19/720) / (A^(n^2/2 + n/2 - 1/3) * exp(7*n^4/32 + 59*n^3/72 + 17*n^2/24 - n/24 + zeta(3)/(8*Pi^2) + zeta'(-3)/6 - 37/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.

A372150 a(n) = Product_{k=1..n} k!^(k^2).

Original entry on oeis.org

1, 1, 16, 161243136, 1953714516870533385423459188736, 18637697331204402735774894643901575833450808531469488619520000000000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Apr 20 2024

Keywords

Crossrefs

Cf. A051675, A255269.

Programs

  • Mathematica
    Table[Product[k!^(k^2), {k, 1, n}], {n, 0, 6}]

Formula

a(n) ~ (2*Pi)^(n^3/6 + n^2/4 + n/12) * n^(n^4/4 + 2*n^3/3 + n^2/2 + n/12 - 1/90) / (A^(1/6) * exp(5*n^4/16 + 5*n^3/9 + n^2/8 - n/12 - zeta(3)/(8*Pi^2) - zeta'(-3)/3 - 13/720)), where A is the Glaisher-Kinkelin constant A074962, zeta(3) = A002117, zeta'(-3) = A259068.
Previous Showing 21-24 of 24 results.

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