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 1-8 of 8 results.

A255322 a(n) = Product_{k=0..n} (k^2)!.

Original entry on oeis.org

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 21 2015

Keywords

Comments

Partial products of A088020. - Michel Marcus, Jul 06 2019

Links

Crossrefs

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

Programs

  • Mathematica
    Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
  • PARI
    {a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

Formula

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .
From Vaclav Kotesovec, Apr 23 2024: (Start)
a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).
a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)

A255438 Decimal expansion of a constant related to A255359.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Feb 24 2015

Keywords

Examples

			6.644987918706354049483118316737842660075362652005201561326290428710371...

Crossrefs

Cf. A255359, A255504, A255511, A255439.
Cf. A074962, A243262, A243263, A243264, A243265.

Formula

Equals limit n->infinity (Product_{k=0..n} (k^4)!) / (n^(1 + 28*n/15 + 4*n^3/3 + 2*n^4 + 4*n^5/5) * (2*Pi)^(n/2) / exp(19*n/9 + n^4/2 + 9*n^5/25)).
Equals 2*Pi*exp(-3*Zeta(5)/Pi^4) * Product_{n>=1} ((n^4)!/stirling(n^4)), where stirling(n^4) = sqrt(2*Pi) * n^(4*n^4 + 2) / exp(n^4) is the Stirling approximation of (n^4)! and Zeta(5) = A013663. - Vaclav Kotesovec, Apr 20 2016

A255439 Decimal expansion of a constant related to A255360.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Feb 24 2015

Keywords

Examples

			11.354954749729782312106630592450215781014...

Crossrefs

Cf. A255360, A255504, A255511, A255438.
Cf. A074962, A243262, A243263, A243264, A243265.

Formula

Equals limit n->infinity (Product_{k=0..n} (k^5)!) / (n^(80/63 + 5*n/2 - 5*n^2/12 + 25*n^4/12 + 5*n^5/2 + (5*n^6)/6) * (2*Pi)^(n/2) / exp(5*n/2 + 35*n^2/144 + n^5/2 + 11*n^6/36)).
Equals 2^(5/4)*Pi^(5/4)*exp(137/3024 - 5*Zeta'(-5)) * Product_{n>=1} ((n^5)! / stirling(n^5)), where stirling(n^5) = sqrt(2*Pi) * n^(5*n^5 + 5/2) / exp(n^5) is the Stirling approximation of (n^5)! and Zeta'(-5) = A259070. - Vaclav Kotesovec, Apr 20 2016

A255511 Decimal expansion of a constant related to A255358.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Feb 24 2015

Keywords

Examples

			4.113740552015338123052453340090368136395763815194771589658140463089224...

Crossrefs

Cf. A255358, A255504, A255438, A255439.
Cf. A074962, A243262, A243263, A243264, A243265.

Formula

Equals limit n->infinity (Product_{k=0..n} (k^3)!) / (n^(29/40 + 3*n/2 + 3*n^2/4 + 3*n^3/2 + 3*n^4/4) * (2*Pi)^(n/2) / exp(n*(n+2)*(12 - 6*n + 7*n^2)/16)).
Equals (2*Pi)^(3/4) * exp(-11/240 - 3*Zeta'(-3)) * Product_{n>=1} ((n^3)!/stirling(n^3)), where stirling(n^3) = sqrt(2*Pi) * n^(3*n^3 + 3/2) / exp(n^3) is the Stirling approximation of (n^3)! and Zeta'(-3) = A259068. - Vaclav Kotesovec, Apr 20 2016

A371624 a(n) = Product_{k=0..n} (n^2 - k^2)!.

Original entry on oeis.org

1, 1, 144, 1755758592000, 66052111513207347990207922176000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 30 2024

Keywords

Comments

The next term has 88 digits.

Crossrefs

Cf. A001142, A255322, A296591, A362288, A371603.

Programs

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

Formula

a(n) = (n^2)!^(n+1) / (A255322(n) * A371603(n)).
a(n) ~ c * A^(2*n) * 2^(4*n*(n^2 + 1)/3) * Pi^(n/2) * n^(4*n^3/3 + n^2 + 5*n/6 + 1/4) / exp(16*n^3/9 + n^2/2 + n), where c = 1.291409... = sqrt(2*Pi) / (A255504 * c from A371603) and A is the Glaisher-Kinkelin constant A074962.

A371642 a(n) = Product_{k=0..n} (n^2 + k^2)! / (n^2 - k^2)!.

Original entry on oeis.org

1, 2, 806400, 29900785676206001356800000, 1118776785681133797769642926006209350326602179759885516800000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A272241, A371624, A371643.

Programs

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

Formula

a(n) = A371643(n) / A371624(n).
a(n) ~ c * 2^(n^2 - n/6 + 1/4) * exp((3*Pi-10)*n^3/9 - n^2 + Pi*n/4) * n^(4*n^3/3 + 2*n^2 + n/2 + 3/4) / A^(2*n), where c = 1.941002... = A255504 * (c from A371603) and A is the Glaisher-Kinkelin constant A074962.

A370483 a(n) = Product_{k=0..n} binomial(n^2 + k^2, k^2).

Original entry on oeis.org

1, 2, 350, 347633000, 101143578356902991250, 422044560230008480282938965899488406272, 1208807563912714402070105775158111317516306396248661153276031151000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A255322, A371643.

Programs

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

Formula

a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2).
a(n) = A371643(n) / ((n^2)!^(n+1) * A255322(n)).
a(n) ~ 2^(4*n^3/3 + n^2 + n/6 + 1/4) * exp((Pi-4)*n^3/3 + Pi*n/4) / (A255504 * n^(n + 1/2) * Pi^(n/2)).

A371644 a(n) = Product_{k=0..n} binomial(n^2 + k^2, n^2 - k^2).

Original entry on oeis.org

1, 1, 10, 57915, 8235313944000, 1077099640691257742845893750, 4629575796245443900868634734946423885068807034000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 31 2024

Keywords

Crossrefs

Cf. A371603, A371624, A371642, A371643, A371645.

Programs

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

Formula

a(n) = A371642(n) / A371645(n).
a(n) = A371643(n) / (A371624(n) * A371645(n)).
a(n) ~ c * exp(Pi*n^3/3 + Pi*n/4 + n) / (2^(2*n^3/3 + 3*n/2) * Pi^(n/2) * A^(2*n) * n^(7*n/6 - 1/4)), where c = 0.761512... = 2^(1/4) * A255504 * (c from A371603) / (c from A371645) and A is the Glaisher-Kinkelin constant A074962.
Showing 1-8 of 8 results.

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