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 1-10 of 10 results.

A216703 a(n) = Product_{k=1..n} (49 - 7/k).

Original entry on oeis.org

1, 42, 1911, 89180, 4213755, 200574738, 9594158301, 460519598448, 22162505675310, 1068725273676060, 51619430718553698, 2496503376570051576, 120872371815599997138, 5857661095679076784380, 284096563140435224042430, 13788153197749122873525936
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, A004994, A049382, A216702, A220607.
Cf. A034835, A386271, A386272, A386273, A386274.

Programs

  • Maple
    seq(product(49-7/k, k=1.. n), n=0..20);
seq((7^n/n!)*product(7*k+6, k=0.. n-1), n=0..20);
  • Mathematica
    Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

Formula

From Seiichi Manyama, Jul 17 2025: (Start)
G.f.: 1/(1 - 49*x)^(6/7).
a(n) = (-49)^n * binomial(-6/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+6). (End)
From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 49^n * Gamma(n+6/7) / (Gamma(6/7) * Gamma(n+1)).
a(n) ~ c * 49^n / n^(1/7), where c = 1/Gamma(6/7) = 1/A220607 = 0.904349... . (End)

A216704 a(n) = Product_{k=1..n} (64 - 8/k).

Original entry on oeis.org

1, 56, 3360, 206080, 12776960, 797282304, 49963024384, 3140532961280, 197853576560640, 12486759054049280, 789163172215914496, 49932506169297862656, 3162392057388864634880, 200447004252955727626240, 12714067126901763295150080, 806919460320698577132191744
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, A203146, A216702, A216703.

Programs

  • Maple
    seq(product(64-8/k, k=1.. n), n=0..20);
seq((8^n/n!)*product(8*k+7, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[64-8/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Sep 23 2017 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 64^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+1)).
a(n) ~ c * 64^n / n^(1/8), where c = 1/Gamma(7/8) = 1/A203146 = 0.917723... . (End)

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)

A248325 Square array read by antidiagonals downwards: super Patalan numbers of order 4.

Original entry on oeis.org

1, 4, 12, 40, 24, 168, 480, 160, 224, 2464, 6240, 1440, 1120, 2464, 36960, 84864, 14976, 8064, 9856, 29568, 561792, 1188096, 169728, 69888, 59136, 98560, 374528, 8614144, 16972800, 2036736, 678912, 439296, 506880, 1070080, 4922368, 132903936, 246105600, 25459200, 7128576, 3734016, 3294720, 4815360
Offset: 0

Views

Author

Tom Richardson, Oct 04 2014

Keywords

Comments

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 4, A025749.

Examples

			T(0..4, 0..4) is:
  1      4      40     480    6240
  12     24     160    1440   14976
  168    224    1120   8064   69888
  2464   2464   9856   59136  439296
  36960  29568  98560  506880 3294720

Links

Crossrefs

Cf. A068555, A025749, A216702 (first column), A034385 (first row), A248324.

Formula

T(0,0)=1, T(n,k) = T(n-1,k)*(16*n-4)/(n+k), T(n,k) = T(n,k-1)*(16*k-12)/(n+k).
G.f.: (x/(1-16*x)^(3/4)+y/(1-16*y)^(1/4))/(x+y-16*x*y).

A386271 Expansion of 1/(1 - 49*x)^(2/7).

Original entry on oeis.org

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

Formula

a(n) = (-49)^n * binomial(-2/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).
a(n) = 7^n * Product_{k=1..n} (7 - 5/k).
In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025

A386272 Expansion of 1/(1 - 49*x)^(3/7).

Original entry on oeis.org

1, 21, 735, 29155, 1224510, 53143734, 2356038874, 106021749330, 4823989594515, 221367522503855, 10227179539678101, 475098976797773601, 22171285583896101380, 1038639455430209672340, 48816054405219854599980, 2300863364299362480145724, 108715793963144877186885459
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A034835, A216703, A386271, A386273, A386274.
Cf. A049381, A216702.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(3/7))

Formula

a(n) = (-49)^n * binomial(-3/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+3).
a(n) = 7^n * Product_{k=1..n} (7 - 4/k).
D-finite with recurrence n*a(n) +7*(-7*n+4)*a(n-1)=0. - R. J. Mathar, Jul 20 2025

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)

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 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.