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.

A216705 a(n) = Product_{k=1..n} (81 - 9/k).

Original entry on oeis.org

1, 72, 5508, 429624, 33832890, 2679564888, 213025408596, 16981168285224, 1356370816782267, 108509665342581360, 8691624193940766936, 696910230823250585232, 55927046023565859464868, 4491372003738673637024784, 360913821729000560118063000
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704.

Programs

  • Maple
    seq(product(81-9/k, k=1.. n), n=0..20);
seq((9^n/n!)*product(9*k+8, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[81-9/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 20 2021 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 81^n * Gamma(n+8/9) / (Gamma(8/9) * Gamma(n+1)).
a(n) ~ c * 81^n / n^(1/9), where c = 1/Gamma(8/9) = 0.927851... . (End)

A216706 a(n) = Product_{k=1..n} (100 - 10/k).

Original entry on oeis.org

1, 90, 8550, 826500, 80583750, 7897207500, 776558737500, 76546504125000, 7558967282343750, 747497875698437500, 74002289694145312500, 7332954160601671875000, 727184620926332460937500, 72159089307305298046875000, 7164366724082454591796875000
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A340725.

Programs

  • Maple
    seq(product(100-10/k, k=1.. n), n=0..20);
seq((10^n/n!)*product(10*k+9, k=0.. n-1), n=0..20);

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 100^n * Gamma(n+9/10) / (Gamma(9/10) * Gamma(n+1)).
a(n) ~ c * 100^n / n^(1/10), where c = 1/Gamma(9/10) = 1/A340725 = 0.935778... . (End)

A216786 a(n) = Product_{k=1..n} (121 - 11/k).

Original entry on oeis.org

1, 110, 12705, 1490720, 176277640, 20941783632, 2495562549480, 298041470195040, 35653210872081660, 4270462368900447720, 512028438031163681628, 61443412563739641795360, 7378329792029068652259480, 886534702703800402679177520, 106574136046464005550646840440
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706.

Programs

  • Maple
    seq(product(121-11/k, k=1.. n), n=0..20);
seq((11^n/n!)*product(11*k+10, k=0.. n-1), n=0..20);
A216786 := proc(n)
    binomial(-10/11,n)*(-121)^n ;
end proc: # R. J. Mathar, Sep 17 2012
  • Mathematica
    Join[{1},FoldList[Times,121-11/Range[20]]] (* Harvey P. Dale, Mar 15 2016 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 121^n * Gamma(n+10/11) / (Gamma(10/11) * Gamma(n+1)).
a(n) ~ c * 121^n / n^(1/11), where c = 1/Gamma(10/11) = 0.942148... . (End)

A248332 Square array read by antidiagonals downwards: super Patalan numbers of order 8.

Original entry on oeis.org

1, 8, 56, 288, 224, 3360, 13056, 5376, 8960, 206080, 652800, 182784, 161280, 412160, 12776960, 34467840, 7311360, 4386816, 5935104, 20443136, 797282304, 1884241920, 321699840, 146227200, 134529024, 245317632, 1063043072, 49963024384, 105517547520, 15073935360, 5514854400, 3843686400
Offset: 0

Views

Author

Tom Richardson, Oct 04 2014

Keywords

Comments

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 8, A025753.

Examples

			T(0..4,0..4) is
  1           8           288         13056       652800
  56          224         5376        182784      7311360
  3360        8960        161280      4386816     146227200
  206080      412160      5935104     134529024   3843686400
  12776960    20443136    245317632   4766171136  119154278400

Links

Crossrefs

Cf. A068555, A025753, A034977 (first row), A216704 (first column), A248324, A248325, A248326, A248328, A248329.

Programs

  • PARI
    matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*64^(n+k)*binomial(n-1/8,n+k)) \\ Michel Marcus, Oct 09 2014

Formula

T(0,0)=1, T(n,k) = T(n-1,k)*(64*n-8)/(n+k), T(n,k) = T(n,k-1)*(64*k-56)/(n+k).
G.f.: (x/(1-64*x)^(7/8)+y/(1-64*y)^(1/8))/(x+y-64*x*y).
T(n,k) = (-1)^k*64^(n+k)*binomial(n-1/8,n+k).

A216787 a(n) = Product_{k=1..n} (144 - 12/k).

Original entry on oeis.org

1, 132, 18216, 2550240, 359583840, 50917071744, 7230224187648, 1028757612985344, 146597959850411520, 20914642271992043520, 2986610916440463814656, 426813850967673556058112, 61034380688377318516310016, 8732611390798600956948971520, 1250010944797171165551838494720
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706, A216786.

Programs

  • Maple
    seq(product(144-12/k, k=1.. n), n=0..20);
seq((12^n/n!)*product(12*k+11, k=0.. n-1), n=0..20);
  • Mathematica
    Join[{1},FoldList[Times,144-12/Range[20]]] (* Harvey P. Dale, Dec 22 2015 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 144^n * Gamma(n+11/12) / (Gamma(11/12) * Gamma(n+1)).
a(n) ~ c * 144^n / n^(1/12), where c = 1/Gamma(11/12) = 0.947376... . (End)

A216788 a(n) = Product_{k=1..n} (169 - 13/k).

Original entry on oeis.org

1, 156, 25350, 4174300, 691890225, 115130533440, 19207610662240, 3210414924974400, 537343198067590200, 90034838076214002400, 15098842345381088202480, 2533860269961226256525280, 425477370330989242241536600, 71480198215606192696578148800
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706, A216786, A216787.

Programs

  • Maple
    seq(product(169-13/k, k=1.. n), n=0..20);
seq((13^n/n!)*product(13*k+12, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[169-13/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Mar 13 2013 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 169^n * Gamma(n+12/13) / (Gamma(12/13) * Gamma(n+1)).
a(n) ~ c * 169^n / n^(1/13), where c = 1/Gamma(12/13) = 0.951742... . (End)
Showing 1-6 of 6 results.

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