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 1-6 of 6 results.

A375839 a(n) = Product_{k=0..n} (k^2 + n).

Original entry on oeis.org

0, 2, 36, 1008, 41600, 2381400, 180457200, 17467670528, 2100621828096, 306960977700000, 53529274174376000, 10973787848179200000, 2611472797582941487104, 713649909809783275801472, 221870902844468552220000000, 77837994361783539267010560000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[k^2 + n, {k, 0, n}], {n, 0, 15}]
    Round[Table[Sqrt[n] * Gamma[1 - I*Sqrt[n] + n] * Gamma[1 + I*Sqrt[n] + n] * Sinh[Sqrt[n]*Pi] / Pi, {n, 0, 15}]]

Formula

a(n) ~ n^(2*n + 3/2) / exp(2*n - Pi*n^(1/2) + 1).

A375840 a(n) = Product_{k=0..n} (k^3 + n).

Original entry on oeis.org

0, 2, 60, 3960, 505920, 111945600, 39501498960, 20891200176000, 15785674348953600, 16407441209402496000, 22748452701706791576000, 41018285140626186366336000, 94161166261926730618189824000, 270252010494895412092926136320000, 954766647796042233397162343121696000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[k^3 + n, {k, 0, n}], {n, 0, 15}]

Formula

a(n) ~ n^(3*n + 2) / exp(3*n - 2*Pi*n^(1/3)/sqrt(3)).

A375841 a(n) = Product_{k=0..n} (k^4 + n).

Original entry on oeis.org

0, 2, 108, 19152, 8840000, 8908817400, 17303456226672, 59111538137501696, 331331804053754904576, 2885800103371503562500000, 37384163240259410286768056000, 694933775143924511454539020849152, 17989643936954432911290280974476623872, 632268529759009258574304284235050340614528
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[k^4 + n, {k, 0, n}], {n, 0, 15}]

Formula

a(n) ~ n^(4*n + 5/2) / exp(4*n - Pi*sqrt(2)*n^(1/4)).

A375843 a(n) = Product_{k=0..n} (k^6 + n).

Original entry on oeis.org

0, 2, 396, 588528, 4087208000, 97390176449400, 6465177278417919600, 1030450155933504769261568, 352805275175791344554903371776, 238031797291547406166218644352900000, 295416986525310718941438520613968960376000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[k^6 + n, {k, 0, n}], {n, 0, 15}]

Formula

a(n) ~ n^(6*n + 7/2) / exp(6*n - 2*Pi*n^(1/6)).

A375844 a(n) = Product_{k=0..n} (k^7 + n).

Original entry on oeis.org

0, 2, 780, 3442680, 94792125120, 11199115535025600, 4424488981755473751120, 4897251346805306604631152000, 13309618365562299179087873337753600, 80202481690565843837334237727510974259200, 987936620325246799505617855507952109963809976000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[k^7 + n, {k, 0, n}], {n, 0, 15}]

Formula

a(n) ~ n^(7*n + 4) / exp(7*n - Pi * n^(1/7) / sin(Pi/7)).

A375845 a(n) = Product_{k=0..n} (k^8 + n).

Original entry on oeis.org

0, 2, 1548, 20400912, 2237404520000, 1316258829530177400, 3107531556500789042401392, 23981023412887138890925360910336, 519343443733819692494314622381817102336, 28055691989665530513724742545624840551562500000
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 31 2024

Keywords

Comments

In general, for m > 2, Product_{k=0..n} (k^m + n) ~ n^(m*n + (m+1)/2) / exp(m*n - Pi * n^(1/m) / sin(Pi/m)).

Crossrefs

Cf. A334411.
Cf. A126804 (m=1), A375839 (m=2), A375840 (m=3), A375841 (m=4), A375842 (m=5), A375843 (m=6), A375844 (m=7).

Programs

  • Mathematica
    Table[Product[k^8 + n, {k, 0, n}], {n, 0, 15}]

Formula

a(n) ~ n^(8*n + 9/2) / exp(8*n - Pi*sqrt(2*(2+sqrt(2)))*n^(1/8)).
Showing 1-6 of 6 results.