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 1-3 of 3 results.

A317980 a(n) = Product_{i=1..n} floor(5*i/2).

Original entry on oeis.org

2, 10, 70, 700, 8400, 126000, 2142000, 42840000, 942480000, 23562000000, 636174000000, 19085220000000, 610727040000000, 21375446400000000, 790891516800000000, 31635660672000000000, 1328697748224000000000, 59791398670080000000000, 2810195737493760000000000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Comments

If p > 2 and p is odd, then Product_{i=1..n} floor(p*i/2) ~ (p/2)^n * n! * 2^(1/(2*p)) * sqrt(Pi) / (Gamma(1/2 - 1/(2*p)) * n^(1/(2*p))).

Crossrefs

Programs

  • Mathematica
    Table[Product[Floor[i*5/2], {i, 1, n}], {n, 1, 20}]
    RecurrenceTable[{4 a[n] - 10 a[n - 1] - 5 (n - 1) (5 n - 6) a[n - 2] == 0, a[1] == 2, a[2] == 10}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)
    FoldList[Times,Floor[5*Range[20]/2]] (* Harvey P. Dale, Sep 17 2020 *)

Formula

a(n) ~ (5/2)^n * n! * 2^(1/10) * sqrt(Pi) / (Gamma(2/5) * n^(1/10)).
Recurrence: 4*a(n) - 10*a(n-1) - 5*(n - 1)*(5*n - 6)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319948 a(n) = Product_{i=1..n} floor(3*i/2).

Original entry on oeis.org

1, 3, 12, 72, 504, 4536, 45360, 544320, 7076160, 106142400, 1698278400, 30569011200, 580811212800, 12197035468800, 268334780313600, 6440034727526400, 161000868188160000, 4347023441080320000, 121716656350248960000, 3651499690507468800000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Floor[i*3/2], {i, 1, n}], {n, 1, 20}]
    RecurrenceTable[{4 a[n] - 6 a[n - 1] - 3 (n - 1) (3 n - 4) a[n - 2] == 0, a[1] == 1, a[2] == 3}, a, {n, 1, 20}] (* Bruno Berselli, Oct 03 2018 *)

Formula

a(n) ~ (3/2)^n * n! * 2^(1/6) * sqrt(Pi) / (Gamma(1/3) * n^(1/6)).
Recurrence: 4*a(n) - 6*a(n-1) - 3*(n - 1)*(3*n - 4)*a(n-2) = 0, with n >= 3. - Bruno Berselli, Oct 03 2018

A319949 a(n) = Product_{i=1..n} floor(4*i/3).

Original entry on oeis.org

1, 2, 8, 40, 240, 1920, 17280, 172800, 2073600, 26956800, 377395200, 6038323200, 102651494400, 1847726899200, 36954537984000, 776045297664000, 17072996548608000, 409751917166592000, 10243797929164800000, 266338746158284800000
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 02 2018

Keywords

Crossrefs

Programs

  • Mathematica
    Table[Product[Floor[i*4/3], {i, 1, n}], {n, 1, 20}]
    RecurrenceTable[{27*(3*n - 7)*a[n] == 54*(2*n - 5)*a[n-1] + 12*(12*n^2 - 42*n + 35)*a[n-2] + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a[n-3], a[1]==1, a[2]==2, a[3]==8}, a, {n, 1, 20}]
    FoldList[Times,Floor[4 Range[20]/3]] (* Harvey P. Dale, Mar 21 2024 *)
  • PARI
    a(n) = prod(i=1, n, (4*i)\3); \\ Michel Marcus, Oct 03 2018

Formula

a(n) ~ (4/3)^n * n! * 2*sqrt(Pi) / (3^(1/4) * Gamma(1/4) * n^(1/4)).
Recurrence: 27*(3*n - 7)*a(n) = 54*(2*n - 5)*a(n-1) + 12*(12*n^2 - 42*n + 35)*a(n-2) + 8*(n-2)*(2*n - 5)*(3*n - 4)*(4*n - 9)*a(n-3).
Showing 1-3 of 3 results.